High-Quality Source of Fiber-Coupled
Polarization-Entangled Photons at 1.56 pm
by
Veronika Stelmakh
Submitted to the Department of Electrical Engineering and Computer
Science
in partial fulfillment of the requirements for the degree of
ARCHIVES
Master of Science in Electrical Engineering
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
at the
JUL 0 1 2012
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2012
o Massachusetts Institute of Technology 2012.
Author .. ..........
All rights reserved.
..........
Department of Electricala
gineering and Computer Sc ce
May 24th, 2012
Certified by............
.....-............
Dr. danco N.C. Wong
Senior Research Scientist
Thesis Supervisor
A*
Accepted by .
Plof. )Si
A.- Kolodziejski
Chairman, Department Committee on Graduate Theses
LBRARIES
2
High-Quality Source of Fiber-Coupled
Polarization-Entangled Photons at 1.56 pm
by
Veronika Stelmakh
Submitted to the Department of Electrical Engineering and Computer Science
on May 24rd, 2012, in partial fulfillment of the
requirements for the degree of
Master of Science in Electrical Engineering
Abstract
This thesis describes the development of a high-quality source of single-mode fibercoupled polarization-entangled photon pairs based on a collinear spontaneous parametric down-conversion process in a bidirectionally pumped periodically-poled potassium titanyl phosphate (PPKTP) crystal inside a polarization Sagnac interferometer
at communication wavelength (1.56 im). A two-photon quantum interference visibility of 97.1% was measured. In addition to the Sagnac source, a linear Mach-Zehnder
source at 810 nm was built for the Interdisciplinary Quantum Information Science and
Engineering (iQuISE) teaching laboratory, to allow students to observe fundamental
quantum phenomena such as the Hong-Ou-Mandel (HOM) dip, the collapse of a Bell
state due to measurement, or the Clauser-Horne-Shimony-Holt (CHSH) inequality
violation in a single experimental setup. An HOM dip with a two-photon quantum
interference visibility of 80% was demonstrated.
Thesis Supervisor: Dr. Franco N.C. Wong
Title: Senior Research Scientist
3
4
Acknowledgments
I would like to thank Dr. Franco Wong for his tremendous help, support, guidance
and patience throughout the past few years. I also would like to thank Prof. Jeffrey
Shapiro and the rest of the Optical and Quantum Communications group. I could
never have asked for a better research group, especially my office mates - post-docs
Maria Tengner and Valentina Schettini. I would like to thank Tian Zhong and UROP
student Rutuparna Das for their help with the detection setup of the Sagnac source,
Nivedita Chandrasekaran for her never ending words of encouragement and proofreading this thesis, and the rest of the group for their fantastic support. Last but
certainly not least, I would like to thank my friends and family, especially my mother
Larissa, my partner Jason and my grandparents Veronika "Senior", Galina and Yuri,
for their incredible support. I would like to dedicate this thesis to my father Nikolai
who passed away so unexpectedly during my second year of graduate school.
5
6
Contents
1
Introduction
15
2
Fundamentals
19
3
4
2.1
Qubits and Entanglement
. . . . . . . . . . . . . . . . . . . . . . . .
19
2.2
Second Order Nonlinear Optical Processes . . . . . . . . . . . . . . .
21
2.3
Nonlinear Crystal Selection and Quasi-Phase-Matching Technique . .
24
2.4
Spatial Emission Modes
. . . . . . . . . . . . . . . . . . . . . . . . .
27
2.5
Classical Interference . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
Sagnac Source Design Considerations, Objectives, and Constraints 31
3.1
Review of Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.2
Design Considerations and Constraints . . . . . . . . . . . . . . . . .
35
3.2.1
Considerations
. . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.2.2
Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
Teaching Laboratory Source at 810 nm
4.1
37
Crystal and Spectrum Characterization . . . . . . . . . . . . . . . . .
38
4.1.1
Second Harmonic Generation
. . . . . . . . . . . . . . . . . .
38
4.1.2
Images of Spontaneous Parametric Down-Conversion Emission
39
4.2
Detection Apparatus
. . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4.3
Sagnac Source at 810nm . . . . . . . . . . . . . . . . . . . . . . . . .
43
4.3.1
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . .
43
4.3.2
Source Characterization
. . . . . . . . . . . . . . . . . . . . .
45
7
4.4
5
Linear Mach-Zehnder Source . . . . . . . . . . . . . . . . . . . . . . .
47
4.4.1
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . .
47
4.4.2
Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
Sagnac Source at 1.56 prn
53
5.1
Experimental Setup . . . . . . . . . . . . . . . .
53
5.2
Detection Apparatus
. . . . . . . . . . . . . . .
58
5.3
Crystal, Filter and Flux Characterization . . . .
61
5.3.1
PPKTP Angle and Temperature Tuning Using Difference Frequency Generation
6
. . . . . . . . . . . .
61
5.3.2
Narrowband Filtering . . . . . . . . . . .
66
5.3.3
Focusing Configuration . . . . . . . . . .
69
5.3.4
Alignment of the Source . . . . . . . . .
71
5.3.5
Flux Characterization
. . . . . . . . . .
73
5.4
Quality of Entanglement
. . . . . . . . . . . . .
76
5.5
D iscussion . . . . . . . . . . . . . . . . . . . . .
81
Conclusion
83
A Calculation of Crystal Grating Period using Sellmeier Coefficients
85
B Loophole-Free Violation of Bell's Inequality
87
8
List of Figures
. . . . . . . . . . . . . .
2-1
The Bloch sphere representation of a qubit.
2-2
(a) Geometry of second harmonic generation. (b) Energy-level diagram
describing energy conservation in second harmonic generation.
2-3
.
.
.
22
(a) Geometry of difference frequency generation. (b) Energy-level diagram describing energy conservation in difference frequency generation.
2-4
20
22
(a) Geometry of spontaneous parametric down-conversion. (b) Energylevel diagram describing energy conservation in spontaneous parametric down-conversion.
2-5
. . . . . . . . . . . . . . . . . . . . . . . . . . .
Geometry of noncollinear propagation of pump kP, signal k8 , and idler
ki with signal emission angle
2-6
23
dr.
. . . . . . . . . . . . . . . . . . . . .
26
A nonlinear crystal such as KTP is periodically poled with period
A.
The crystal is composed of evenly spaced ferroelectric domains
with alternating polarizations along the direction of light propagation.
Quasi-phase matching is achieved by choosing the period and finetuning it by changing the temperature of the crystal.
2-7
. . . . . . . . .
26
(a) Phasor diagram for the superposition of two waves intensities I1
and I2 and phase difference
(
= 02 -
p1. (b) Dependence of the total
intensity I on the phase difference o [60].
9
. . . . . . . . . . . . . . .
29
3-1
Sagnac interferometer polarization scheme for type-II down-conversion.
Pairs can be generated in the clockwise (Ev) and counterclockwise
(EH) directions.
The orthogonally polarized outputs are separated
at the polarizing beam splitter (PBS) to yield polarization-entangled
. . . . . . . . . . . . . . . . . . . . . .
36
4-1
Second harmonic generation experimental setup . . . . . . . . . . . .
38
4-2
Second harmonic generation output emission with 3C bandwidth FWHM. 39
4-3
Collinear and non-collinear SPDC emission.
. . . . . . . . . . . . . .
39
4-4
Experimental setup to see the SPDC rings. . . . . . . . . . . . . . . .
40
4-5
Images of SPDC emission taken using a Princeton Instruments Ver-
signal and idler photon pairs.
sArray camera from collinear (upper left) to ring-like (lower right) as
a function of crystal temperature, from 12'C to 28'C as labeled. . . .
41
4-6
Coincidence apparatus designed by Kim [39]. . . . . . . . . . . . . . .
42
4-7
810nm Sagnac interferometer setup. . . . . . . . . . . . . . . . . . . .
43
4-8
Measurement of classical interference between the signal and idler paths
in the Sagnac interferometer . . . . . . . . . . . . . . . . . . . . . . .
45
Sagnac source setup for Hong-Ou-Mandel dip experiment.
. . . . . .
46
4-10 Linear source setup for the Hong-Ou-Mandel dip experiment. . . . . .
48
4-9
4-11 Coincidence dip observed during one demonstration of the teaching
laboratory as a function of distance translated. . . . . . . . . . . . . .
50
4-12 Beamsplitter with input modes a and b and output modes a' and b'. .
51
5-1
Experimental Setup of the Sagnac Source at 1560 nm. Focal lengths
are given in cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
5-2
Schematic of the self-differencing setup [56].
. . . . . . . . . . . . . .
59
5-3
Efficiency of detectors 1 and 2 cooled at -20
C versus gating voltage.
60
5-4
Dark counts registered for detectors 1 and 2 cooled at -20'C versus
5-5
gating voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
Difference frequency generation experimental setup. . . . . . . . . . .
61
10
5-6
Comparaison of 3 different crystals by difference frequency generation.
The Raicol crystal with grating period 46.1 prm was most efficient and
crosses the degenerate wavelength line, meaning that the optimum temperature of efficient SPDC generation occurs at the degenerate wave-
length. .........
5-7
63
...................................
Measured signal wavelength as a function of crystal temperature and
tilting angle characterization for the 1 cm, 46.1
tm grating period
R aicol crystal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-8
64
Measured signal wavelength as a function of crystal temperature and
tilting angle characterization for the 2.5 cm, 46.2 pum grating period
A dV R crystal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-9
64
Theoretical model for degenerate collinear emission for fixed temperatures (a) and for fixed grating periods (b).
. . . . . . . . . . . . . . .
65
5-10 Calculated SPDC emission for 3 cm crystal with a grating period of
46.1 p m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
5-11 Calculated SPDC emission with calculated Bragg filter shape for the
2.5 cm crystal (red curve), 3 cm crystal (green curve), and 5 cm crystal
(blue curve). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
5-12 1.8 nm Optigrate Bragg filter efficiency normalized to total SPDC output: 97.54% for the 2.5 cm crystal (red curve), 97.78% for the 3 cm
crystal (green curve), 97.43% for the 5 cm crystal (blue curve). ....
5-13 Bragg grating characterization at fixed angle varying the wavelength.
5-14 Focusing geometry in the crystal with Rayleigh range
ZR,
68
69
crystal length
L, beam-waist radius wo, and focus offset zo. . . . . . . . . . . . . . .
70
5-15 (a) and (b) Simultaneous optimization of the total collection probability P 2 and the symmetric heralding ratio qi. In (a) the collection
probability is plotted along the left axis and the heralding ratio is plot-
ted along the right axis [13]. . . . . . . . . . . . . . . . . . . . . . . .
11
70
5-16 Spontaneous parametric down conversion observed with a Goodrich
InGaAs camera at 35'C. The emission was focused to match the pixel
size FW HM (30 pm).
. . . . . . . . . . . . . . . . . . . . . . . . . .
73
5-17 Measured singles counts with subtracted dark counts (21 000 dark
counts/sec and 9 250 dark counts/sec for detectors 1 and 2, respectively) for signal, idler as a function of pump power (mW) operating
at -20'C with 10% duty cycle. . . . . . . . . . . . . . . . . . . . . . .
76
5-18 Measured coincidence counts with subtracted dark counts (21 000 dark
counts/sec and 9 250 dark counts/sec for detectors 1 and 2, respectively) for signal, idler as a function of pump power (mW) operating
at -20'C with 10% duty cycle. . . . . . . . . . . . . . . . . . . . . . .
77
5-19 Average fringe visibility recorded: 94.97% in the horizontal and vertical
(HV) basis and 94.99% in the diagonal and anti-diagonal (AD) basis
without subtracting accidentals. . . . . . . . . . . . . . . . . . . . . .
12
79
List of Tables
.
74
.
78
A.1
Table of Sellmeier coefficients used for all phase matching calculations
85
A.2
Calculated indices of refraction at T= 25'C
. . . . . . . . . . . . . .
86
5.1
Optical component transmissions and overall system efficiencies
5.2
Measured singles and coincidences in two mutually unbiased bases.
13
.
.
14
Chapter 1
Introduction
In quantum information science, information is stored in quantum systems such as
photons, ions, atoms, nuclear spins, or quantum dots. As these quantum systems
interact with each other and the environment, the information is processed by the
laws of quantum mechanics.
Unlike classical digital states, a two-state quantum
system, or qubit, can be in a superposition of the two states at any given time.
Entanglement, a correlation beyond what classical physics allows, is an essential part
of quantum information science. Historically, entanglement is the unique quantum
property that explains the Einstein-Podolsky-Rosen (EPR) paradox [26] and was used
to formulate Bell's inequality [6] [7] to distinguish quantum mechanics from hiddenvariable theories.
More recently, entangled states have been used to create secure
quantum key distribution (QKD) systems [27], perform quantum computing using
linear optics [25] [41] [40], transport quantum states from one system to another [10]
[21], potentially increase channel capacities in communication lines [11], and image
at higher resolutions than possible with classical light [64], among other applications.
For efficient quantum operations, these applications generally require a steady and
large supply of highly-entangled quantum states.
Spontaneous parametric down-
conversion (SPDC) is a second-order nonlinear frequency mixing process [20] in a
non-linear crystal which splits an input photon (pump) into two photons (signal and
idler) whose frequencies sum to the input frequency.
SPDC has been frequently
utilized for the generation of two-photon or multi-photon entangled states.
15
Since
the first demonstration of high efficiency polarization entanglement by Kwiat et al.
in 1995 [47], the performance of SPDC-based entanglement generation has shown
tremendous improvement.
Communicating information in an energy efficient or spectrally efficient manner at
high rates for long distances is the goal of most communication systems. The photon's
long coherence time and the ease with which it can be coupled into a single-mode fiber
make it an ideal candidate for exchanging quantum information between remote sites.
As a result, it has been considered an essential quantum resource for transferring a
quantum state through a quantum network [21], establishing entanglement over long
distances on the order of kilometers [65] [53] [52] [25], and is utilized to transmit
information secretly in quantum key distribution (QKD) [9]. The primary concern
when building a source of entangled photons, is to ensure that the source is efficient
and bright. Simply increasing the power of the pump laser at the crystal is not a
good way to ensure efficiency. As the pump power is increased, more photon pairs are
generated per second. However the likelihood of having more undesirable multi-pair
events which can compromise the security of some communication protocols is also
increased. Instead it is best to optimize the source efficiency by ensuring that the
least amount of pairs are wasted. To achieve this, it is critical that the emission is
optimally coupled into a single-mode fiber.
The goal of this thesis is to develop a high-flux high-quality source of fiber-coupled
polarization-entangled photon pairs based on a collinear SPDC process in a bidirectionally pumped periodically-poled potassium titanyl phosphate (PPKTP) crystal
inside a polarization Sagnac interferometer at communication wavelength (1.56 Pm).
The outputs of the source, the signal and idler, are coupled into their respective
single-mode fibers. The efficiency of the single-mode coupling is optimized by pump
focusing while still retaining high conditional probability for pair detection. Such a
source allows convenient and efficient transport of entangled photons for a variety of
applications, from QKD to linear-optics quantum computing.
16
Organization
The second chapter of this thesis reviews some fundamental concepts in quantum
and classical optics. The third chapter of the thesis focuses on the design objectives,
considerations and constraints of a Sagnac source and provides a summary of key entanglement sources built in the past. As an introduction to our main experiment, the
fourth chapter discusses the preparatory experiments that were made in the course
of building a teaching laboratory for the Interdisciplinary Quantum Information Science and Engineering (iQuISE) program. The fifth chapter then examines the most
important aspect of the thesis - the design and characterization of the Sagnac source
at 1.56 prm that was built as a joint research project with Lincoln Laboratory. In the
final chapter a summary of the main accomplishments of this thesis is provided, as
well as a discussion on further improvements to the Sagnac source and its potential
applications.
17
18
Chapter 2
Fundamentals
2.1
Qubits and Entanglement
In classical computation and classical information, the bit is a fundamental unit that
represents the amount of information stored by a physical system. The classical bit
can take one of two separate states at a time, generally represented by 1 and 0. The
quantum bit, or qubit, is the quantum analog of the bit and can be represented in
either state 1l) or 10). The difference between bits and qubits is that a qubit can be
in a linear combination of the two states that form the computational basis. This is
referred to as a superposition and can be written as:
|1) ,
(2.1)
are complex numbers and I12 and
1 2 represent the probability of
1P) = aI0) +
where a and
#
being in the 10) state or the 11) state, respectively, while satisfying the total probability
theorem:
Ia 2 +
||2=1.
(2.2)
Geometrically, we can visualize the qubit state as a unit vector in a two-dimensional
19
complex vector space and we can rewrite Eq. 2.1 as:
')
where 0 and
#
=
9
0
cos 0) + e' sin l1),
(2.3)
define a point on the unit three dimensional sphere, often called the
Bloch sphere and shown in Fig. 2-1. Measuring a qubit projects its state onto one
of the states in its computational basis by collapsing it from its superposition state
(Eq. 2.1) to either 1) or |0). This behavior does not have a classical equivalent and
is one of the fundamental postulates of quantum mechanics.
lo) Az
0
: 1)
Figure 2-1: The Bloch sphere representation of a qubit.
Entanglement, or as Einstein called it "spooky action at a distance", is another
phenomenon that cannot be explained classically. A multi-partite pure quantum state
is said to be entangled when it is not possible to express the joint state as a product
state, thus measuring one state clearly affects the outcome of the other state. If
entangled, one object cannot be fully described without considering the other object.
Entangled states can be implemented via a number of different physical systems such
as photons and ions. In our experiment with photons we are using entanglement in
polarization but photonic entanglement exists in other degrees of freedom such as
momentum, time-bin, and energy-time. In our experiment, photons are generated by
spontaneous parametric down-conversion, a process that is described in detail in the
following section.
20
2.2
Second Order Nonlinear Optical Processes
The linear relationship between the electric polarization of a dielectric medium and
the electric field of a light wave is usually described by:
(2.4)
P = coXE
where P is the polarization, co is the electric permittivity of free space (8.854 x 10-12
F m-1 in SI units), x is the electric susceptibility of the medium, and E is the
electric field. However this is an approximation that is only valid when the electric
field amplitude is small. The more general form of Eq. 2.4 is a nonlinear relationship
between the polarization and electric field:
P = cox('1 E +
2 +
±ox(
COX(3)E
...
(2.5)
The first term of this equation corresponds to Eq. 2.4 and describes the linear response
of the medium. The other terms of this equation describe the nonlinear response of
the medium. We can write:
P(l) = coXy1E
p(
2
)
= COX
p(
3
)
-=o
2
)E
(2.6)
2
)E
(2.7)
(2.8)
(2.9)
P(n) =oX(n)E"
where for n > 2 P(n) is the n-th order nonlinear polarization and X()
(2.10)
is the n-th
order susceptibility.
In the experiments described in this thesis, we have used three common X( second
order nonlinear optical processes.
21
1. Second Harmonic Generation (SHG): the input frequency is doubled (w -> 2w).
(a)
(b)
0)
2o)
0)
Figure 2-2: (a) Geometry of second harmonic generation. (b) Energy-level diagram
describing energy conservation in second harmonic generation.
2. Difference Frequency Generation (DFG): the output frequency is the difference
between two input frequencies (w1 , W2
-±
(a)
W3
wi
-
W2 ).
(b)
()3
Figure 2-3: (a) Geometry of difference frequency generation. (b) Energy-level diagram
describing energy conservation in difference frequency generation.
3. Spontaneous ParametricDown-Conversion (SPDC): an input photon is split into
two photons whose frequencies sum to the input frequency. It is "stimulated" by
vacuum fluctuations and converts only a small fraction of the input pump light
(P 3 -- W1 ,W2 with wi + W2
= U 3 ).
22
(b)
(a)
---
1------r-CD1
31)3(2)
2
(0 2
(- 3
(0
2
Figure 2-4: (a) Geometry of spontaneous parametric down-conversion. (b) Energylevel diagram describing energy conservation in spontaneous parametric downconversion.
By energy conservation, the signal photon and idler photon produced at the output
have their sum frequencies equal to the pump frequency. They must also obey the
law of momentum conservation which is achieved by a phase-matching technique
described in the next section.
23
2.3
Nonlinear Crystal Selection and Quasi-PhaseMatching Technique
The interaction of an incident optical beam with a dielectric medium, such as a crystal,
can be described as a process in which the electromagnetic field of the optical beam
induces a dipole polarization in the dielectric medium. This causes the radiation of
a new electromagnetic field. We already saw that polarization P of the material is
described in terms of this electromagnetic field by Eq. 2.5. However due to the well
defined axes of crystalline materials, we need to consider the directions in which the
fields are applied. The second-order nonlinear polarization p( 2 ) is then rewritten as:
(2.11)
E
eoZXiijkEk
j,k
where the subscripts i, j, and k correspond to the crystal's axes in Cartesian coordinates x, y, and z. This nonlinear response can be rewritten with the nonlinear optical
and so
coefficient tensor dij using the fact that Xz(2) EEz must be equal to X(2
forth.
diX = dijkk =
(2.12)
(n)
where the index follows the convention of 1=11, 2=22, 3=33, 4=23, 5=13, and 6=12.
The components of the second-order nonlinear polarizations can therefore be written
as Eq. 2.13:
dnl d 12 d 13 d 14 d 15 d16
P
P(2)
Pz
-
d 21
d 22
d 23
d 24
d 25
(2.13)
d 26
d31 d 32 d 33 d 34 d 35 d 36
2
z
\2Exy
24
In many crystals the nonlinear optical coefficient tensor can be simplified because the
crystal symmetry requires many of the terms to either equal each other or be zero.
Eq. 2.13 illustrates that depending on the polarization of the involved electromagnetic
fields, different d coefficients govern the nonlinear interaction.
Common crystals used for SPDC include
#-barium
borate (BBO), periodically
poled lithium niobate (PPLN), and periodically poled KTiOPO 4 (PPKTP). By varying the phase matching conditions, the emission of the crystal can be tuned. The
phase matching condition can be interpreted as momentum conservation in the limit
of a long interaction length. The energy conservation equation can then be written
as w, = w,
+ wi where w is the angular frequency and p, s, and i refer to the three
participating modes in SPDC generally labelled pump, signal, and idler. In the limit
of long interaction lengths, the momentum conservation becomes:
k, = k, + ki
(2.14)
where kP, ks, and ki are the pump, signal, and idler wave vectors, respectively. The
equation for collinear propagation can also be rewritten in terms of index of refraction
of the crystal n and wavelength in vacuum A:
nP
AP
ni
n.
2=- +
.
As
Ai
(2.15)
A number of different phase matching techniques exist such as phase matching
by temperature tuning or by angle tuning. For bulk crystals the angle of emission
is set by the angle phase-matching condition and results in non-collinear emission if
it is not zero. For periodically poled crystals, such as PPKTP in our experiment,
the phase matching condition in Eq. 2.14 is replaced with a quasi-phase matching
condition which depends on the poling period A of the crystal and the quasi-phase
matching order m:
Ik, -
ks - kil = 2Am
A
25
(2.16)
By poling a crystal with a user-defined period and fine-tuning the period by placing
the crystal in a temperature-controlled oven, we can phase-match at any desirable
operating wavelengths within the crystal's transparency window.
x
Z
y
Figure 2-5: Geometry of noncollinear propagation of pump kp, signal k8 , and idler ki
with signal emission angle
#,.
periodically poled crystal
t
pump
beam
k,
-_,
k
' k,
'
period
A
ferroelectric domains
periodically inverted
Figure 2-6: A nonlinear crystal such as KTP is periodically poled with period A. The
crystal is composed of evenly spaced ferroelectric domains with alternating polarizations along the direction of light propagation. Quasi-phase matching is achieved by
choosing the period and fine-tuning it by changing the temperature of the crystal.
26
2.4
Spatial Emission Modes
In order to be able to transport our generated SPDC light, it is most convenient to
couple it into a single-mode fiber, which requires optimizing the SPDC output for
single-mode collection. To understand the behavior of the SPDC, we must look at
the full quantum mechanical expression describing the joint state of the signal and
idler in terms of spatial distribution [58]. The evolution of the quantum state in the
Schr6dinger picture is:
IT) = exp -i-
JO+
H(t) dtl |Woo)
(2.17)
which can be approximated by the first two-terms of a perturbative expansion:
1 --
) =
H(t) dt
|Roo)
(2.18)
hto
where IToo) is the state at time to, T is the interaction time, and H(t) is the interaction
Hamiltonian given by:
H(t)
j
x(+)E-)E
)d3r + H.c.
(2.19)
JV
where V is the volume of the crystal that is illuminated by the pump laser field Ep+),
x is the nonlinear electric susceptibility tensor, and H.c. is the Hermite conjugate.
We simplify by assuming a classical pump field with a constant intensity given by:
E+)
Ei(kpr- wt)
(2.20)
The output fields of the signal and idler can be written as:
E-=
where at
j
d 3rat,
~e(kjr-wjt)
(2.21)
is the creation operator of a photon for frequency wj and polarization
mode k and
j
can represent either signal or idler fields. Assuming that all fields are
27
propagating along the z-axis, for a type-II process with continuous-wave monochromatic plane wave pump we can obtain the orthogonal signal and idler fields state by
carrying out the integration of Eq. 2.21 in one dimension over the crystal length L:
|'(wi, o)) =
J
dwi dw 3(wp - wi - w,) sinc (LAk)
at
) ,v(ws) 4 oo)
(2.22)
The spectral amplitude sinc(Lk) with the phase mismatch Ak = k, - ki - k, comes
from integration of the interacting electric fields over the finite crystal length L. The
energy uncertainty caused by the finite interaction time in the crystal is negligible in
regard to the phase matching, and is therefore represented by a 6 function.
28
Classical Interference
2.5
Our Sagnac source is an interferometer that must be properly aligned for generating
high quality polarization entanglement.
The first step in aligning an interferome-
ter is to maximize its classical interference.
In general, when two equal-frequency
monochromatic waves with complex amplitudes E 1 (r) and E2(r) are superposed, the
result is a monochromatic wave of the same frequency that has a complex amplitude E(r) = E1(r)
12
+ E 2 (r). The corresponding intensities are I1 = |E1(r)12 and
= |E 2 (r)l2 and the intensity of the total field is:
I = |E
2
=
(2.23)
|E1 + E212 = El12 + IE212 + E*E2 + E1E2*
where E 1 and E2 can be written as E1= v71 exp(ip1) and E2
= v'12
exp(ip 2 ) when
omitting the explicit dependence on r for convenience. Then the total intensity can
be written as:
I = I1 + 12 + 2
where p =
02 -
4Io(cos2 ( /2))
01.
If Ii
=
12
I1
(2.24)
cos y
= Io then Eq. 2.24 yields I
=
21o(1 + cos 0) =
so that for a = 0, and I = 4Io which corresponds to maximum
constructive interference. For o = 7r the total intensity I is zero, which corresponds
to destructive interference.
U
U2
(b)
(a)
-47r -21r
0
2ir
47r
SO
Figure 2-7: (a) Phasor diagram for the superposition of two waves intensities I1 and
12 and phase difference P =
02 -
V1.
(b) Dependence of the total intensity I on the
phase difference p [60].
29
The PPKTP crystal used in the experiments in this thesis has type II emission.
It produces a collinear signal and idler that are orthogonally polarized to each other.
In our experiments we used a probe laser polarized at 450 to create a horizontally
polarized beam and a vertically polarized beam after a polarizing beam splitter to
simulate the signal and idler emission. Since interference can only occur when two
beams have the same polarization, we needed to find a way to rotate the beams to
have the same polarization and to vary their phase without having to separate them
in order to measure classical interference. This was achieved by rotating them both
by 45' in polarization to switch to the diagonal and anti-diagonal (AD) basis instead
of the horizontal and vertical (HV) basis, and projecting them along the horizontal
orientation after their passage through a polarizing beam splitter. Since the horizontal beam and vertical beam are orthogonal in polarization, their indices of refraction
in PPKTP are different with different rates of tuning as a function of temperature.
Thus the temperature of the crystal can be varied to induce a relative phase shift
between the two collinear beams to yield the classical interference for aligning the
interferometer.
The relationship between wavelength, temperature and index of refraction can be
written as Atempi = 9(n
2
- ni) for one temperature and
IA
4
temp2 =
2 -
n)
for another temperature. Subtracting one from the other we then have:
A/temp2
-
Aitempi =
2 -
n
--
n 2 + ni) = 2-r
(2.25)
where L is assumed to be constant, due to very small variation. At a wavelength of
1.56 ptm a shift of about 3'C results in a full period from one constructive interference
to the next.
30
Chapter 3
Sagnac Source Design
Considerations, Objectives, and
Constraints
We began our design by considering the other sources of polarization-entangled photons that have been built over the years. A number of sources have been built using
different crystal types. Earlier sources of polarization entangled light were made using BBO crystals, in a single or double bulk crystal configuration [47] [48].
More
recent sources use the periodic poling technique described in Section 2.3 of the previous chapter on bulk crystals such as PPLN [42] and PPKTP [30] or waveguides [70].
Other materials used for periodic poling are wide band gap inorganic crystals like
lithium tantalate [54], and some organic materials [16].
Furthermore, a number of
different configurations were tested over the years. In the next section, a summary
of the most relevant sources built is provided, including the type of crystal used, the
type of configuration, the pump wavelength, the source brightness (the number of detected coincidences per unit time per unit pump power per detection bandwidth) and
the quantum-interference visibility (when available). Furthermore, design objectives
and constraints for our Sagnac source are discussed.
31
3.1
"
Review of Sources
In 1995, Kwiat et al. [47] generated polarization-entangled photon pairs by collecting the photons at the intersection of two emission cones from SPDC in a single
,3-barium borate (BBO) nonlinear crystal with type-II phase-matching to generate non-collinearly propagating photon pairs that were polarization entangled and
obtained a normalized flux of 0.07 pairs/s/mW/nm with a high degree of entanglement, 97.8% two-photon quantum interference visibility. This method used a
relatively simple setup but had some limitations in generating high-flux output
mainly due to the non-collinear phase-matching condition of the BBO crystal and
the need to use several types of filtering to have indistinguishable spatial, spectral, and temporal modes. Non-collinear geometry also prevented the use of a long
crystal for increasing the flux and the system required degenerate signal and idler
output wavelength.
" To resolve some of the issues in the first source, in 1999 Kwiat et al. [48] then combined two orthogonally-polarized output modes from two cascaded BBO crystals
with type-I phase matching. With the improved overlap of the signal and idler
cones in this configuration, Kwiat et al. achieved 0.24 pairs/s/mW/nm with a visibility of 99.6% for a small aperture or 28 pairs/s/mW/nm with a lower visibility of
90% for a larger aperture. Compared to the single BBO system, the signal polarization at each point of the output cone was entangled with the idler polarization at
the opposite point on the cone and hence more entangled pairs were expected from
the entire cone. In practice, however the maximum output flux was still limited by
the non-collinear output modes and could not be collected efficiently, thus limiting
the useful output flux and preventing the use of long crystals. Furthermore, this
arrangement still required small apertures as it suffered from the problem of spatial
distinguishability.
" In 2004, Kuklewicz et al. [44] used a periodically poled 1 cm KTiOPO 4 (PPKTP)
crystal in a collinear phase-matched configuration to increase the output flux. This
setup required post-selection and the same types of filtering as the single BBO
32
setup. This experiment yielded a higher spectral brightness of 300 pairs/s/mW/nm
after timing compensation and post selection with a 50:50 beam splitter, which was
an order of magnitude larger than the double BBO system designed by Kwiat et
al. However similar to the BBO systems, the interference between the signal and
idler output from the type-II SPDC system required timing compensation due
to crystal birefringence, and this also required a small aperture to avoid spatialmode distinguishability, and therefore the effective output flux was limited at high
visibility.
* In 2004, Fiorentino et al. [29] coherently pumped a single PPKTP crystal from
two opposite directions and combined the two output modes with a Mach-Zehnder
interferometer (MZI) in order to make the output indistinguishable in all other
modes except the polarization. A detected flux of -12
000 pairs/s/mW was ob-
tained in a 3 nm bandwidth with a two-photon quantum interference visibility of
90%. This setup increased the output flux by removing the requirements for spatial,
spectral, and temporal filtering, but the fidelity of the output was limited mainly
due to the need for phase stabilization of the MZI geometry which was sensitive to
environmental perturbation, and the large size of the interferometer.
* In 2006, Kim et al. designed a phase-stable source configuration by embedding
the bi-directionally pumped PPKTP crystal inside a polarization Sagnac Interferometer [30] [67] (pump wavelength -405 nm, signal and idler -810
nm, 10 mm
PPKTP crystal with a 10.03 pm grating period for frequency-degenerate type-II
quasi-phase-matched collinear parametric down-conversion). A normalized flux of
700 pairs/s/mW/nm with quantum interference visibility of 99.45% was obtained.
In this configuration, the 405 un pump was weakly focused to a beam waist of
~160 pm at the center of the crystal. Compared to initial BBO experiments, the
flux was increased by five orders of magnitude in pairs/s/mW/nm, mainly due to
the fact the PPKTP crystal allows a collinear output mode.
The entanglement
quality was also improved by using the phase-stable Sagnac interferometer and the
double SPDC configuration.
33
* In 2007, Fedrizzi et al. demonstrated a fiber-coupled, wavelength-tunable source
of narrowband, polarization entangled photons in a Sagnac configuration (pump
wavelength ~405 nm, signal and idler ~810 nm, 25 mm type II PPKTP bulk
crystal) [28]. A spectral brightness of 273 000 pairs/s/mW/nm was obtained, a
factor of 28 better than comparable previous sources, with a quantum interference
visibility of 98.7% and a system efficiency of 28.5% with a focused pump beam.
Wavelength tuning of entangled photons in a range of ±26 nm around degeneracy
showed virtually no decrease in entanglement.
* In 2012, White et al. designed a fiber-coupled source of polarization entangled
photons in a Sagnac configuration (pump wavelength -410 nm, signal and idler
-820 nm, 10 mm type-IL PPKTP bulk crystal) [63]. A focusing of 200 pm for the
pump spot diameter and 84 pum collection mode diameter in the crystal were used.
With these parameters, a typical pair detection efficiency of 40% was achieved.
This was measured with standard single-photon avalanche diodes (SPADs), whose
detection efficiency were estimated to be 50% at 820 nm, implying a collection
efficiency of 80%. The high heralding efficiency in this setup provides a path to a
photonic loophole-free Bell test.
34
3.2
Design Considerations and Constraints
The final goal of the thesis was to design a high-flux high-quality source of fiber
coupled polarization-entangled photons at 1.56 pm.
The considerations and con-
straints considered in the process of designing the source are listed below.
When
optimized, a single-mode fiber-coupled PPKTP based Sagnac source should produce
a high entangled-pair collection efficiency with very high polarization entanglement,
and thus was chosen as our final design configuration.
3.2.1
Considerations
The Sagnac configuration was chosen for its many advantages. The intrinsic phase
stability of the Sagnac-type setup, the compactness, and the ease of use, and reported
high visibilities by our group [28] and others [63] made the bi-directionally pumped
Sagnac configuration (Fig. 3-1) our first choice. Another advantage of the Sagnac is
that degenerate emission is not required for polarization entanglement in a Sagnac
interferometer and the signal and idler are automatically separated. The Sagnac configuration is symmetric and timing compensation is not required. This configuration
is more compact and more immune to vibrations than other interferometers such
as the Mach-Zehnder interferometer (MZI). Bell states can be easily produced in a
Sagnac source due to having precise control over the output phase.
PPKTP was chosen as it is commonly used for type-I and -I
phase-matched
second harmonic generation for pump wavelengths of 730-3500 nm. One advantage
of PPKTP is that it produces a collinear degenerate emission when appropriately
phase matched by simply tuning the temperature.
Single-mode coupling was chosen for several reasons. The detectors used in this
experiment require single-mode fiber coupling. Furthermore, research by Bennink [13]
and verified by White [63] suggests that single-mode coupling can lead to a high
conditional probability. Indeed, the efficiency of the single-mode coupling can be
optimized by pump focusing while still retaining high conditional probability for pair
detection, to be discussed further in Chapter 5.
35
Finally coupling into single-mode
fiber allows convenient and efficient transport of entangled photons over relatively
long distances for a variety of applications.
3.2.2
Constraints
One constraint of the Sagnac source is that standard angle tuning phase matching
techniques cannot be used because the pumping of the crystal is bidirectional and
thus needs to be symmetric, thus the quasi-phase matching technique is used in
combination with temperature tuning to produce collinear outputs.
Another constraint of our Sagnac source is that single-mode fiber coupling of
the SPDC emission outputs is needed for InGaAs single photon detectors that are
single mode fiber-coupled.
For single-mode fiber collection of the SPDC outputs,
it is necessary to have precise alignment of the source output to ensure that the
counterclockwise (CCW) and clockwise (CW) emission are spatially overlapped as
shown in Fig. 3-1, indistinguishable and can both be coupled into the single mode
fiber.
Counterclockwise Photon Pair Production
Clockwise Photon Pair Production
Idler
Idler
EH
Ey
Mirror 1
DM
/
EH
Mirror 1
Signal
PBS
PPKTP
DM
Signal
PBS
PPKTP
Dual-A HWP
Mirror 2
Dual-A HWP
Mirror 2
Figure 3-1: Sagnac interferometer polarization scheme for type-II down-conversion.
Pairs can be generated in the clockwise (Ev) and counterclockwise (EH) directions.
The orthogonally polarized outputs are separated at the polarizing beam splitter
(PBS) to yield polarization-entangled signal and idler photon pairs.
36
Chapter 4
Teaching Laboratory Source at
810 nm
The purpose of the Teaching Laboratory was to give first hand experience in experimental quantum optics to students in the interdisciplinary quantum information
science and engineering (iQuISE) program by allowing them to observe fundamental
quantum phenomena such as the Hong-Ou-Mandel (HOM) dip, the collapse of a Bell
state due to measurement, or the Clauser-Horne-Shimony-Holt (CHSH) inequality
violation, in one experimental setup. The first source built for the laboratory was a
Sagnac source at 810 nm based on another source previously built in our group [30].
However the Sagnac configuration was not versatile enough to achieve the goal of
performing the three experiments in one setup. It is not straightforward to go from
measuring an entangled state to looking at the HOM dip in a Sagnac configuration,
as explained in Section 4.3.
As a result we switched to constructing a linear sys-
tem. The following sections describe the experiments that were performed in order
to understand and prepare us for the design and characterization of a source of polarization entangled photons at communication wavelength. These sections describe the
successes as well as the failures encountered in the process of building the teaching
laboratory. The final Sagnac source built for MIT Lincoln Laboratory is described in
the next chapter.
37
4.1
4.1.1
Crystal and Spectrum Characterization
Second Harmonic Generation
As previously described, second harmonic generation (SHG) is a process in which
the input frequency is doubled (w --+ 2w).
This is ideal for confirming PPKTP
nonlinearity and phase matching wavelengths for SPDC, a process in which an input
photon is split into two photons (W3
-+
W1 , 02 with wi
+W 2 =
W3 ), as it can be seen
as the reverse process for degenerate operation wi = w2 .
HWP2
HWP1
810 nm
pump laser
foj
b
Dual-)
PBS
IF
f1PPT
405 nm
output
Figure 4-1: Second harmonic generation experimental setup
In the SHG measurement, an 810.0 nm beam polarized at 450 was sent into the
PPKTP crystal.
The confocal parameter b = 2nirw 2/A where n is the refractive
index of the crystal for wavelength A and wo is the pump beam waist, was set equal
to the length of the crystal. The output emission was detected and measured through
a lock-in amplifier, using an interference filter. After optimizing the beam inside the
crystal and finding the optimum temperature for the emission, we found the PPKTP
nonlinear coefficient to be deff
~ 2.5 pm/V for the 2 cm crystal in line with the
expected value. Using the probe laser at 810.0 nm to generate SHG at 405.0 nm we
found the optimum operating temperature to be T = 31.8'C i 1.5'C with bandwidth
AT = 3C FWHM, as shown in Fig. 4-2.
38
1.0
0
0.8
0
U)
0.6
M
0.4
E
0
z
0.2
00
|
25.0
Figure 4-2:
-I27.0
29.0
33.0
31.0
Temperature ('C)
35.0
37.0
Second harmonic generation output emission with 3C bandwidth
FWHM.
4.1.2
Images of Spontaneous Parametric Down-Conversion
Emission
An effective experimental technique to confirm SPDC emission is to image the emission from the crystal using a high sensitivity Princeton Instruments VersArray camera
(Model 7381-0001) while changing the temperature of the crystal. As the temperature changes, the index of refraction of the crystal changes, and the emission angles
of the signal and idler beams change as well.
---~ ~~-~ ~ - -- - --
------- ------------------- - ---------------------- ----------------------------
non-collinear SPDC
collinear SPDC
Figure 4-3: Collinear and non-collinear SPDC emission.
39
We were able to go from a collinear state to a non-collinear ring-like state by
changing the temperature by about 15 C. In this setup (Fig. 4-4) the PPKTP was
pumped by a 405 nm pump beam which was then filtered using dichroic mirrors
(DM). The beam was then collimated in order to maximize the efficiency of the 1 nm
bandpass filter and focused again. The camera was placed after the focus so that
the ring-like state of the emission could be observed. By tuning the temperature and
changing the k-vectors of the emission we observed variation from the non-collinear
SPDC state to the collinear SPDC state.
Fourier Plane
f2 = 6cm
fl= 20cm
PPKTP
DM
DM
'/''
I nm bandpass filter
Figure 4-4: Experimental setup to see the SPDC rings.
40
Figure 4-5: Images of SPDC emission taken using a Princeton Instruments VersArray
camera from collinear (upper left) to ring-like (lower right) as a function of crystal
temperature, from 12'C to 28'C as labeled.
41
4.2
Detection Apparatus
For both the Sagnac experiment and the HOM dip experiment in the teaching laboratory, Perkin Elmer SPCM-AQRH Si avalanche photodiodes (APD) single photon
counting modules were used in conjunction with a homemade coincidence counting
board. These relatively low cost APDs have high efficiencies (> 50%) for wavelengths
around 800 nm with low dark counts (-500/s), can be coupled to multi-mode and
single-mode fibers, and do not require any additional cooling equipment making them
ideal detectors for a student laboratory. The coincidence counting apparatus designed
by T. Kim [39] was connected to a National Instruments SCB- 68 card. A program
written in Labview allowed us to look at single counts in each arm, coincidence counts
from the board, and to calculate the conditional detection efficiency.
Lab made
coinc. counter
ay
re
SPCM TTL
signal
photons
output
PECL TTL
Voltage
TTL
output
idler
photons
split
30s
HL
30-m
l-m
PECL PECLAND
PT
TTL output
coinc. counts
_4
Perkin
Elmer
68 pin
delay
SPCM
co ne cto
cabl
single
counts
info
Figure 4-6: Coincidence apparatus designed by Kim [39].
Taking into account the transmission efficiencies of optical components, coupling
efficiencies, and detection efficiencies, the overall signal and idler detection efficiencies were estimated to be -10%.
The coincidence board allowed a variable time
coincidence window by using different lengths of electrical delay cables. In our experiments, the coincidence window was set at ~2 ns. Since the number of dark counts
was low and pumping power was low, accidentals were believed to be minimal and
not subtracted in any of the measurements.
42
4.3
4.3.1
Sagnac Source at 810nm
Experimental Setup
The first source that we built was a bi-directionally pumped Sagnac interferometer
with 405 nm pump and degenerate signal and idler at 810 nm. The chosen flux grown
Raicol PPKTP crystal was 2 cm long with a grating period of 10.1 Pm for frequencydegenerate type-II quasi-phase-matched collinear parametric down-conversion with
anti-reflection coatings at 405 and 810 nm. The temperature of the crystal could be
maintained from 10 to 50'C to within t0.01'C by a homemade temperature controller
using the PID1500 1.5 A temperature controller from Wavelength Electronics. Optical
components and detection equipment for this wavelength were readily available which
was one of the deciding factors in designing the source at this wavelength for the
student lab.
----------------------------------------Single
photon
detector
I
405 nm pump laser
IF
IF
Detection
Apparatus
PBS
810 QWP2
fiber
adaptor
DualA80HP
QWP
Dual-A
HWP1
Idler
r-
- --
-
- --
II Mirror 1
l1 S1
MirrI1
1
Bi-directionally
pumped Sagnac
configuration
- -4- -- Io
DM fi
Q
0 :
0 a
o 0
Single
photonI
IF detector
~~~Signal
nnrnr1r--
/7+
S
I
Coincidence
Detector
IU
I
PBS
1
PPKTP
Sagnac
*12
S2
HWP2
IPBS
-- - - - - - - - - - - - -
I
Mirror 2
1 -I
Figure 4-7: 810nm Sagnac interferometer setup.
43
-- -
In the setup shown in Fig. 4-7, a 405.2 nm continuous-wave Toptica laser was
coupled into a single-mode fiber for spatial mode filtering and easy transport.
A
half-wave plate and a quarter-wave plate transformed the fiber-coupled pump light
into the appropriate polarization. At the PPKTP output, before detection, a spectral bandwidth of 1 nm was imposed by using an interference filter (IF) centered at
810.0 nm with a maximum transmission of 66% at the center wavelength. This filter
in addition to dichroic mirrors ensured that no pump was leaking through to the
detectors. The polarization states of the signal and idler photons were analyzed with
a combination of a HWP, QWP and a PBS before coupling into single-mode fibers.
The signal and idler outputs were then sent to Perkin Elmer Si avalanche photodiode
single-photon counting modules with a photon detection efficiency of 50% at 810 nm
and coincidence counting was done using a homemade coincidence counting apparatus [39]. A more thorough description of the generation of entanglement in a Sagnac
source is provided in Chapter 5.
In order to generate maximally entangled states, the clockwise and counterclockwise paths of the Sagnac interferometer must be indistinguishable. One proven way
to achieve high spatial overlap between the two paths is to align the interferometer
by looking at the classical interference between the two paths.
As previously ex-
plained in Section 2.5, the PPKTP crystal has type II emission, producing a collinear
signal and idler that are orthogonally polarized to each other. Sending an 810 nm
probe laser polarized at 450 through the signal arm, we create a horizontally polarized beam and a vertically polarized beam after the polarizing beam splitter (PBS)
that propagate counterclockwise and clockwise, respectively, when the dual halfwave
plate (HWP2) is set to zero. In order to measure the classical interference between
these two orthogonally polarized beams at the output of the PBS in the idler arm,
we rotate them both by 45' in polarization to switch to the diagonal (D) and antidiagonal (AD) basis instead of the horizontal and vertical (HV) basis, and project
them along the horizontal orientation after their passage through a second polarizing beam splitter. Since the horizontal beam and vertical beam are orthogonal in
polarization, their indices of refraction in PPKTP are different with different rates
44
of tuning as a function of temperature.
Thus the temperature of the crystal can
be varied to induce a relative phase shift between the two collinear beams to yield
interference between the signal and idler paths for aligning the interferometer.
A
classical fringe visibility of 95.8% in the transmitted output port of the analyzer PBS
and 95.7% in the reflected port was obtained, where classical interference visibility is
defined as (Imax
-
Imin)/(Imax + Imin).
2.50000
2 .00000
-- P Trans (mW)
I .:UUUU
+
P Refi (mW)
1.00000
0500
0.00000
0
100
200
Angle (*)
300
400
Figure 4-8: Measurement of classical interference between the signal and idler paths
in the Sagnac interferometer.
4.3.2
Source Characterization
Pumping the crystal with -1 mW of 405 nm pump, and collecting the emission
into multi-mode fiber we saw -200 000 singles/s detected and -25 000 coincidence
counts, with a coincidence window of -2 ns, corresponding to a conditional probability of 12.5%. While this moderate system efficiency was satisfactory for the student
laboratory, one of the main goals of the teaching laboratory was to have a versatile
experimental setup that would allow students to see the HOM dip as well as perform
a CHSH Bell's inequality violation. We attempted to do so with the setup shown in
Fig. 4-9 but the alignment proved to be difficult and thus good spatial overlap and
high classical interference were hard to achieve.
45
405 nm
pump laser
fiber
adaptor
Dual-A
HWP1
Detection
Apparatus
V
-j
Idler
Mirror
----------
--------
------------------------fl
I
HWP
Signa
0
IF
PBS
PPKTP
Sagnac
HWP
PBS
Mirror
- -- -- -- -- -- - --
50:50 Bc
I
PBS IF
~
I
Figure 4-9: Sagnac source setup for Hong-Ou-Mandel dip experiment.
A number of issues were encountered with this setup. Visually, for students who may
have never seen optical setups before, the design was very convoluted. This was in
part due to the use of a disk 50:50 beamsplitter. This component made the alignment
very difficult as it shifted the beams slightly due to its thickness. With the additional
HOM setup, a higher classical visibility than ~90% was never reached. This indicated
that a high spatial overlap was difficult to achieve and therefore a high visibility for
two-photon quantum interference would be even harder to achieve since the beams
need to be nearly indistinguishable. The same issues applied to entanglement, as
it was difficult to ensure that the bidirectionally pumped Sagnac had high classical
interference visibility. The Sagnac setup alone could be optimized for entanglement,
however looking at both entanglement and the HOM dip while still coupling into the
fibers connected to the detectors was significantly harder in the Sagnac setup than in
the linear Mach-Zehnder setup that we built instead, described in Section 4.4.
46
4.4
Linear Mach-Zehnder Source
As previously stated, the purpose of the teaching laboratory was to give first hand experience in experimental quantum optics to students in the interdisciplinary quantum
information science and engineering (iQuISE) program by allowing them to observe
fundamental quantum phenomena such as the Hong-Ou-Mandel (HOM) dip, the collapse of a Bell state due to measurement, or the Clauser-Horne-Shimony-Holt (CHSH)
inequality violation, in one experimental setup. In our first attempt at designing the
HOM dip experiment we tried to build on the existing Sagnac source design. While
a Sagnac source alone could have been optimized for a high degree of entanglement,
the additional alignment to look at the HOM dip proved to be difficult and created
a convoluted design difficult to visually understand for students, especially ones with
no prior knowledge of optics. The Sagnac implementation of the system lacked versatility, as explained in Section 4.3.2. While a Sagnac interferometer provides a number
of advantages over a Mach-Zehnder interferometer to generate polarization entangled
photons, as described previously in Section 3.2, easily performing three experiments
in the Sagnac setup was difficult. For that reason it was decided that the redesign
of a visually simpler source would aid the students in their effort to comprehend the
setup and perform the CHSH and HOM dip experiments.
4.4.1
Experimental Setup
We chose a simple linear system for the source in a Mach-Zehnder (MZ) interferometric configuration (Fig. 4-10). In this setup, a 405 nm diode laser pump generated
signal and idler from SPDC in a 2 cm type-IL PPKTP crystal. The vertically polarized idler was reflected at the polarizing beam splitter (PBSO) and the horizontally
polarized signal was transmitted at PBSO and sent into symmetric arms of the MZ
interferometer with identical path lengths. A second polarizing beam splitter (PBS1)
ensured that the horizontal polarization of the signal was as pure as possible at the
50:50 beam splitter (BS), provided that half-wave plate HWP1 was set at 0' (if HWP1
was rotated by 450, the polarization of the signal became vertical). In the other arm,
47
the vertical polarization of the idler was rotated by 900 by HWP2 and PBS2 ensured
that the now horizontal polarization of the idler going into the 50:50 BS was as pure
as possible resulting in two spatially indistinguishable input photons at the 50:50 BS.
Idler
0
810nm
Probe Laser
Detector 1
IF
PBS4
HWP4
Prism2
SM Fiber
rHWP1
MirrorPBS1
7
QPHWPo
L
s
s
f~i
IF
50:50 BS
1
fu
PBS2
HWP2
DM
405nm
Laser
HWP3
PBS3
PS
Signal
Dtco
Mirror
Mirror
Mirror
PKP
PPKTP
VIPim
f PBSO
Pim
Mirror
Figure 4-10: Linear source setup for the Hong-Ou-Mandel dip experiment.
In order to observe the HOM dip or obtain a polarization entangled state, a high
degree of spatial overlapping of the modes of the SPDC emission must be ensured.
To do so, a measurement of the classical interference in the MZ interferometer was
made to check the alignment. In our setup, the classical interference measurement
was made by sending a probe at 810 nm at 450 polarization. The transmitted and
reflected probe beams would have the same polarization at the 50:50 BS and could
interfere with each other. The interference was then measured after the 50:50 BS
through an additional side arm with a second polarizing beam splitter. The optical
path of one of the beams was slowly translated while the maximum and minimum
amplitude of its interference signal was observed at one of the outputs of the PBS. A
classical visibility of 98% was observed using Thorlabs Si detectors connected to an
oscilloscope.
48
The next step in the alignment was to match the pump beam to the probe beam.
The assumption when optimizing the alignment of the interferometer using a probe
laser is that the SPDC would also follow the same optimized path and a high level
of entanglement would be achieved if no optical components that would affect the
signal or idler (probe) paths were touched.
To match the pump to the probe, a
Thorlabs beam profiler was used to overlap the pump beam and probe beam at two
different locations (one right at the crystal, and one about 50 cm away from the
crystal, using an additional lens to refocus the beam). By going back and forth and
adjusting the pump mirrors, one mirror for the near-position and another mirror for
the far-position, we were able to overlap the two beams visually within the resolution
of the beam profiler.
The final alignment was made by pumping the crystal and
looking at the SPDC emission in the multi-mode fiber using the APDs. Adjusting
the two mirrors in front of the analyzer PBS and tuning the IR filter allowed us to
optimize the coincidence counts, thus insuring that the correct photon pairs were
getting coupled into the fiber. Pumping the crystal with -1 mW of 405 nm pump,
and collecting the emission into multi-mode fiber we saw ~250 000 singles/s generated
and -19 000 coincidence counts, with a coincidence window of -2 ns, corresponding
to a conditional probability of ~7.6%.
4.4.2
Measurements
Once the setup was aligned and coincidence counts were detected, the setup needed
to be optimized for the HOM dip experiment.
The HOM dip tests the degree of
indistinguishability of the two incident photons [35] and therefore their polarizations
must be the same at the 50:50 beam splitter to interfere with each other. This is
easily done by rotating HWP1 in the signal path.
When HWP1 is set to 00 the
horizontally polarized signal beam is unaffected and can interfere with the horizontal
idler beam (provided that HWP2 was initially set correctly to rotate the vertically
polarized idler beam by 90' in polarization).
Looking a the coincidence counts, we
slowly translated the prism in the signal arm until a dip in coincidence was observed.
49
The Hong Ou Mandel dip experiment allows us to calculate the coherence time of
the interfering photons. Two indistinguishable photons entering the two ports of a
beamsplitter at the same time do not exit through two different output ports - they
always exit through the same one. When two photodetectors monitor the outputs
of the beamsplitter, the coincidence rate of the detectors will drop to zero when the
identical input photons overlap perfectly in time. As the optical path difference of
one of the photons is varied through translation of the prisms, a dip in coincidence
corresponding to the maximum interference between the two photons is observed
(Fig. 4-11).
3
1500
1400
5
1300
0
e
1200
1100
$
.
'U
1000
900
800
0
700
0
600
500
-5
-3
-1
1
3
5
Optical Path Difference (mm)
Figure 4-11:
Coincidence dip observed during one demonstration of the teaching
laboratory as a function of distance translated.
In this initial measurement, the visibility of the dip was not optimized. After spending
more time on tuning the temperature to fine tune the collinear emission, cleaning up
the polarization with additional quarter-wave plates, and adding pinholes in front of
the multi-mode fiber for increased spatial selection, a visibility of ~80% was observed.
The full width at half maximum (FWHM) coherence time r of the observed HOM
dip can be calculated as:
r
cal
3.645 ps ,
50
(4.1)
where A corresponds to the wavelength of the emission (810 nm), c is the speed of
light and Al is the distance translated.
One of the main advantages of implementing the linear Mach-Zehnder setup shown
in Fig. 4-10 in the teaching laboratory was that it allowed us to switch easily between
implementing the HOM dip demonstration and creating entangled photon pairs in
the same experimental setup by simply rotating the correct halfwave plates. Entanglement was created in the following manner. First, rotating HWP1 by 450 ensured
that the two incoming photons at the 50:50 BS were orthogonally polarized and did
not interfere with each other. The vertically polarized photon has a 50% chance of
going to detector 1 or detector 2 and the horizontally polarized photon also has a
50% chance of going to detector 1 or detector 2. The photons are in a superposition
state after they pass through the beamsplitter. We do not know through which port
the photon has passed.
a'
U
a
Figure 4-12: Beamsplitter with input modes a and b and output modes a' and b'.
The relationship between the input and output modes for a 50:50 beamsplitter can
be described as:
ft=
bt =
b't)
(4.2)
(d't + ib't)
(4.3)
1(id't
51
+
After the beamsplitter, the post-selection state in which one photon is in path 1 and
one photon is in path 2 has two possibilities: both photons are transmitted or both
are reflected. Because the two possibilities are equally likely and the two processes
are coherent, the output state under post-selection is:
1
|T-) =
(IH)IV) 2 - IV) 1 1H) 2 ),
This state is called a singlet state and is one of the Bell states.
(4.4)
Bell states are
maximally-entangled bipartite states that exhibit stronger-than-classical and nonlocal
correlations when their joint statistics are measured by two parties and can be written
in the horizontal/vertical basis in the following form:
I-)
=
(IH)1|V)2
-
IT+)
=
(IH)1IV) 2
+|V
<D-)
=
(IH)I|H)2 -
<D+)
=
(IH)I|H)2 + dV) 1 |V) 2 ).
V) 11H) 2 ),
(4.5)
)1 |H) 2 ),
(4.6)
IV) 1 |V) 2 ),
(4.7)
(4.8)
A commonly used method for characterizing the entanglement quality is the
measurement of two-photon quantum interference visibility. To compare the quality of the fringe pattern, we obtain the quantum-interference visibility defined as
(Cmax - Cmin)/(Cmax
+ Cmin), where Cmax is the maximum number of coincidence
counts and Cmin is the minimum number of coincidence counts. This measurement
must be performed in more than one basis to ensure that we are indeed observing an
entangled state. In the horizontal/vertical (HV) basis, the analyzer HWP of one arm,
signal for example, is set to H(0 0 ) or V(90') and the HWP on the other arm, idler,
is scanned for all angles while recording the coincidence counts, resulting in a fringe
pattern. The same procedure is then repeated in the diagonal/anti-diagonal (AD)
basis, fixing the signal arm's analyzer HWP to D(+450 ) or A(-450 ) and scanning the
idler's analyzer HWP. Here, quantum visibility of 95% and 86% were observed in the
HV basis and the AD basis, respectively.
52
Chapter 5
Sagnac Source at 1.56 pm
The main goal of this thesis was to design a high-flux source of single-mode fibercoupled high-quality polarization entangled photons at telecom wavelength. In this
chapter, the experimental setup of the source is described, taking into consideration
the lessons learned while building the iQuISE teaching laboratory. The detection
apparatus was characterized and the novel technique of self-differencing is briefly
described in this chapter as well. The source characterization is also provided, from
crystal grating period and SPDC bandwidth calculations, to choice of narrowband
filtering, focusing configuration, source alignment and finally flux and entanglement
characterization.
Based on lessons learned while testing the sources implemented for the iQuISE
laboratory, changes were made to the design of the Sagnac source discussed in this
chapter. Extra degrees of freedom on the crystal mount were removed for additional
stability. A new PBS mount was machined as it played a crucial role in the alignment
of the beam and needed to be extremely stable.
A description of the techniques
implemented to help us align the source by measuring classical interference is also
provided.
5.1
Experimental Setup
53
5.
C)
CD
C)
0
(D
Signal
PPKTP
To Detector 1
..
..
...
...................
.....
. .........
..
....
.........
....
..
....
Idler
To Detector 2
In the experimental setup shown in Fig. [5-1], a 780.2 nm continuous-wave Toptica
DL Pro Laser was coupled into a single-mode fiber for spatial mode filtering and easy
transport into the Sagnac source. A half-wave plate (HWP1) and a quarter-wave plate
(QWP1) transformed the fiber-coupled pump light into the appropriate elliptically
polarized light of Eq. [5.1] to provide the power balance and phase control of the
pump field:
Z
where
IH and iv
=
EHZH + e'*PEyiv
(5.1)
are the horizontal (H) and vertical (V) polarization unit vectors, and
<D, is the relative phase between the horizontal and vertical components. The horizontally polarized pump was transmitted through the dual wavelength PBS while the
vertically polarized pump was reflected. Thus the 3 cm long PPKTP crystal was bidirectionally pumped. The counter-clockwise (CCW) beam arrived at the crystal with
the horizontal polarization needed to generate SPDC. However the clockwise (CW)
propagating beam was vertically polarized.
Therefore a dual wavelength halfwave
plate (dual HWP) was used to rotate the pump beam to a horizontal polarization
necessary for SPDC generation.
The CW generated pair then arrived at the PBS
without any additional rotation of the polarization. In contrast the CCW generated
pair was rotated by 900 in polarization. Thus the CCW generated two-photon state
'PH)
and CW generated two-photon state |Wv) after the PBS can be written as:
IqH) =C4H E H|Hs)1|Vi)2
(5.2)
|py) = ei*rvEylVs)il Hi) 2
(5.3)
where <DH and <DV are the accumulated phases through their respective optical paths
and components,
T
1H and TIV are the generation efficiencies, and EH and EV are the
magnitudes of the H and V components of the pump. Subscript s and i refer to signal
(path 1) and idler (path 2) respectively. The final superposition state can therefore
55
be written as:
|IJ) cXes*g(COS 0|H,)1|Vi)2+ e"' sin 0|V)1|Hi)2)
where
4
(5.4)
represents the global phase, <D, represents the relative phase that is fully
adjustable by varying the pump phase <D,, and 0 = tan-
.v~)
Therefore, a
singlet state was easily generated by simply adjusting the pump HWP1 and QWP1
to set 0 to 450 and D, to -F.
The flux grown Raicol PPKTP crystal was 3 cm long with a grating period of
46.1 pm for frequency-degenerate type-II quasi-phase-matched collinear parametric
down-conversion with anti-reflection coatings at 780 and 1560 nm. The temperature
of the crystal was maintained by a Thorlabs TED200C temperature controller to
within ±0.01'C for temperature ranging from 20'C to 60'C. The tight pump focusing of -27 pm was chosen to maximize the SPDC coupling into single mode fiber.
This corresponds to signal and idler beam waists of 38 pm. The reasons for choosing
this tight focus are discussed in more details in Section 5.3.3. Silver mirrors were
used inside the interferometer in order to maximize the reflection efficiency for both
the pump and SPDC wavelengths (63.4% at 780 nm 96.7% at 1560nm). While dielectric mirrors are generally more efficient when designed for the correct wavelength
and provide better polarization maintainance we were unable to find ones that were
highly reflective at both wavelengths. The dichroic mirrors outside the interferometer were specifically designed to reflect 99.9% of the light at 1560 nm and transmit
97% of the light at 780 nm. This attenuation of the pump ensured that the 90 dB
attenuation by the reflective Bragg grating was enough to prevent any pump from
leaking into our detectors. A spectral bandwidth of 1.8 nm was imposed by using
the Bragg grating centered at 1560.5 nm with a maximum reflection of 99% at the
center wavelength. Even though it would have been easier to use Bragg gratings in
transmission, their peak efficiency of 92% would reduce the overall detection efficiency
which is undesirable. The use of reflective Bragg gratings added a level of complexity
to the setup in terms of alignment because the filter needed to be angle-tuned to
match the SPDC output to the peak of the filter and the fiber coupling of the SPDC
into the single-mode fiber required adjustment after each angle tuning step.
The 1 cm dual wavelength Bernhard Halle Nachfl. polarizing beamsplitter inside
the Sagnac interferometer was characterized to have an extinction ratio of less than
1:500 at 780 nm and 1560 nm in transmission and less than 1:20 in reflection. These
numbers were not as good as we had hoped and certainly should be improved in
any future configuration to ensure that the polarization is a pure as possible, thus
maximizing the quality of the entanglement.
Zoom lenses were added to the signal and idler arms in the setup to optimize
the coupling into single-mode fibers and ensure that the beam was collimated when
incident to the Bragg grating. The simplest implementation of a zoom lens uses 3
f
and one concave lens with
in between the two convex lenses.
In this setup, two convex
lenses: two identical convex lenses with focal length
a focal length -f/2
lenses with
f
= 10 cm and one concave lens with
f
= -5
cm were used rather than
purchasing a packaged zoom lens. The zoom lenses allowed us to change the size of
the beam waist, while fixing the location of the beam waist, by translating two of the
three lenses.
Finally, the polarization states of the signal and idler photons were analyzed with
a combination of a HWP, QWP and a PBS before being coupled into the single
mode fiber. The signal and idler outputs were then sent to indium gallium arsenide
(InGaAs) APD single photon counters and coincidence counting was done using a
PicoQuant Hydraharp (multichannel picoscond event timer). The detection setup is
described in detail in Section 5.2.
57
5.2
Detection Apparatus
For this experiment, Princeton Lightwave PGA-300-CUS indium gallium arsenide
(InGaAs) APDs single photon counters were used in conjunction with pulse detection
circuitry designed by NIST. The detection system was developed jointly by our group
at MIT and NIST [69]. The Princeton Lightwave InGaAs APD is used as a single
photon avalanche diode (SPAD) that is optimized to generate a macroscopic electrical
pulse in response to an incident photon. SPADs are biased above their breakdown
voltage in order to operate in a regime known as the Geiger-mode in which a single
photon can trigger a self-sustained electron-hole generation avalanche process that
gives rise to a macroscopic current pulse. A typical operation mode is to increase
the bias above the breakdown for a short duration with a gating pulse. During this
period, the SPAD is armed for single photon detection [14].
Typically, the applied
bias between gating pulses is kept at a steady-state waiting mode below breakdown.
However, during the avalanche, some carriers can be trapped in deep levels. These
trapped carriers decay and can re-trigger an avalanche during the subsequent gating
occurrences [37] in an effect commonly referred to as afterpulsing. The afterpulsing
effect limits the repetition rate at which one can arm and operate the SPAD [8]. The
dark count rate is the average rate of registered counts without any incident light.
This determines the minimum count rate at which the signal is primarily caused by
real detection events. The false detection can also be of thermal origin and can be
strongly suppressed by cooling the detector below room temperature.
The NIST pulse detection circuitry uses the concept of self-differencing to minimize noise [68]. This minimization is achieved by splitting the signal, delaying one
branch by one cycle, and subtracting the resulting signal from the un-delayed branch.
This way, any noise present through all cycles will cancel out, whereas detections
(which are not present in all cycles) will remain. The delay is created mechanically
by using a longer cable in one branch than in the other. The length of the delay is
fine tuned using phase-trimmers.
58
Bias-tee
NB
Mechanical delay
Optical fiber from laser source
AT
De
s s-Mechanical
+
adjustment
+
To oscilloscope and/or
comparators
A
Figure 5-2: Schematic of the self-differencing setup [56].
In our detection setup, the synthesizer frequency was set at 628.5MHz which
corresponds to a 1.6 ns period. The 600 ps square gate was open ~150 ps per cycle and
yielded a duty cycle of -10%.
The efficiency and dark counts of our detectors were
characterized at different temperatures ranging from room temperature to -30'C.
Lowering the temperature causes a decrease in the dark count rate. However, it also
becomes more difficult to maintain the cooler temperature and afterpulsing increases.
An operating temperature of -20'C provided a good compromise between relatively
high efficiency (Fig. 5-3) and relatively low dark counts (Fig. 5-4). The thermoelectric
cooler (TEC) was able to maintain the temperature using only an additional exterior
fan without water cooling. For the characterization of the Sagnac source the detectors
were operated on average at 20% efficiency in order to minimize both afterpulsing
effects and dark counts. At these operating conditions, the breakdown voltages for
detectors 1 and 2 were found to be 68.5V and 67.9V, respectively. A gating voltage
of 70.5V and 70.1V was applied and we observed 21 000 dark counts/sec and 9 250
dark counts/sec for detectors 1 and 2, respectively. This variation falls within the
normal range of variation for different detectors.
59
100
U
L.
0
90
80
70
60
50
40
30
20
10
0
-p
4
-
-*-Detector 1
-rA
69
69.5
-U-Detector 2
70
70.5
71
71.5
Gate Voltage (V)
Figure 5-3: Efficiency of detectors 1 and 2 cooled at -20'C versus gating voltage.
90,000
*)
0
80,000
70,000
60,000
50,000
40,000
30,000
-+-Detector 1
-E-Detector 2
20,000
10,000
0
69
69.5
70
70.5
71
71.5
Gate Voltage (V)
Figure 5-4: Dark counts registered for detectors 1 and 2 cooled at -20'C versus
gating voltage.
60
5.3
Crystal, Filter and Flux Characterization
PPKTP Angle and Temperature Tuning Using Differ-
5.3.1
ence Frequency Generation
A standard way to characterize a nonlinear crystal used for parametric down conversion is to first test it by means of another frequency-mixing process such as difference frequency generation (DFG). As previously discussed the output frequency
in difference frequency generation is the difference between two input frequencies
(w1 , W2
-+
W3
=
Wi
-
W2 ). In our experimental setup (Fig. 5-5), we used a tunable
wavelength Ti-Sapphire laser as the pump, and a tunable Agilent infrared laser as
the idler. Polarizing beam splitters (PBS) were utilized in order to ensure that the
horizontal and vertical polarizations of our beams were maximized.
Horizontally Polarized Pump Beam
780 n n
pump la ser
Chopper
PBS
L2i
InGaAs
Photodiode
Detector
fi
4,
I
Filter
f3
PBS
o~bi
PBS
f2
1560 n m
idler lascer
QWPHWP
Dichroic
Mirrors
PBS
V
Mirror
Vertical ly Polarized Idler Beam
To Lock-in
Amplifier
1i
Mirror
7
Dichroic
Mirror
PPKTP
Mirror
Horizontally Polarized Signal Beam
Figure 5-5: Difference frequency generation experimental setup.
The 780 nm and 1560 nm beams were configured to have a focused beam waist
at the center of the crystal. After the crystal the 780 nm light was filtered out using
dichroic mirrors and an interference filter. While the first PBS ensured that the polarization of the beam was as pure as possible, the last two polarizing beamsplitters
before the detector provided additional filtering by ensuring that the vertically polar-
61
ized idler exited through the vertical output port and were not transmitted. Finally,
the horizontally polarized signal was detected by a Thorlabs InGaAs photodetector
and measured using a lock-in amplifier synchronized to the optically chopped pump
beam. The nonlinear coefficient of the crystal deff can be determined from the DFG
measurement [49]:
deff
c Eo
n n ni A (1/k, + 1/kj)Ps
P
16
P Pp 16 TrL hm
(5.5)
where c is the speed of light, so is the dielectric constant, nP, no, ni are the indices of
refraction of the pump, signal and idler respectively, AP, A8, Ai are the wavelengths
of the pump, signal and idler respectively with corresponding wave vectors k,, ks, ki.
Pp, Ps, P are the pump power, signal power and idler power, respectively. L is the
length of the crystal and hm is the reduction factor for a focusing parameter (= L/b
where b = 2n7w/A, where wo is the pump beam waist. The calculated deff should
be comparable to the one given by the manufacturer.
Crystals from different vendors with different grating periods and lengths (Raicol
1 cm crystal with a grating period of 46.1 pm, Raicol 0.5 cm crystal with a grating
period of 46.21 pm, and AdVR 2.5 cm crystal with a grating period of 46.2 Pm) were
tested using the DFG setup described above to ensure that the maximum emission
could be tuned to be collinear and degenerate by appropriately tuning the temperature
and tilting the incident angle of the crystal. The further the operating temperature
is away from room temperature, the more difficult it is to maintain the temperature
with a small heatsink. In addition, cooler temperatures (as high as 10'C) can induce
condensation on the crystal, depending on the humidity of the room. After taking
these issues into account, the optimum temperature resulting in collinear emission for
our crystal was determined to be 35'C. The Raicol crystal with a grating period of
46.1 pm was chosen because it was the most efficient crystal and a collinear degenerate emission could be reached within the operating temperature range of 18'C to
65'C imposed by our crystal oven thermoelectric cooling and heating capabilities. In
addition, it showed the largest deff of 2.75 pm/V in our tests as indicated in Fig. 5-6.
62
1600
1590
1580
E
Grating Period (um)
Length (cm)
d eff (x 10A-12)
Pump Wavelength (nm)
Degenerate Emission (nm)
Temperature (*C)
Ralcol
46.21
0.50
2.14
767
1534
24
AdVR
46.20
2.50
1.78
774
1548
24
Raicol
46.10
1.00
2.75
787
1574
24
1570
--- AdVR 46.2um
-0-Raicol 46.2um
*Raicol 46.1um
-Degenerate
1560
1550
1540
1530
765
770
775
780
785
790
795
8 )0
Pump (nm)
Figure 5-6: Comparaison of 3 different crystals by difference frequency generation.
The Raicol crystal with grating period 46.1 pam was most efficient and crosses the
degenerate wavelength line, meaning that the optimum temperature of efficient SPDC
generation occurs at the degenerate wavelength.
By changing the temperature of the crystal by about 15'C, we were able to move
the peak emission wavelength by about 1 nm. Tilting the crystal away from the
normal incidence angle also shifted the maximum emission wavelength by as much as
1 nm for both the Raicol (Fig. 5-7) and AdVR (Fig. 5-8) crystals. Take for example,
the Raicol 46.1 pm temperature tuning curve of Fig. 5-7 (left panel). For a pump
wavelength of 780.405 nm (solid squares), the degenerate wavelength is 1560.81 nm,
located somewhat between the two measured points near the temperature of 32'C.
The data suggests that a higher temperature shifts the vertically polarized idler (along
the crystal's z-axis) to a longer wavelength. Also, at a given crystal temperature, a
shorter pump wavelength shifts the idler to a shorter wavelength.
63
Such information as in Fig. 5-7 helped us tune the operating wavelengths of the
Sagnac source during the experiment.
Tilting Angle and Temperature Dependence for
two pump wavelengths at 24 0C and 400C
(Raicol Crystal 46.1Opm 1.0cm)
Temperature versus Wavelength
(Raicol Crystal 46.1Opm 1.0cm )
1562
7
y = 0.0875x + 1558.1 -
1561.5
'A
0
0
-+-779.927nm
temp=24.1 C
-0-779.927nm
temp=40.1 C
5-
I..
0* 5
S
0
1561
E
1560.5
6.
-+-779.927
I
1560
0
0* 3
nm pump
-0-780.405 nm pump
1559.5
,-r-780.405nm
temp=24.OC
0*
2++-780.405nm
temp=40.OC
1
y = 0.05x + 1558.2
1559
24
29
34
1559 1560 1560 1561
39
1561 1562 1562 1563
Wavelength with max emission (nm)
Temperature of crystal (*C)
Figure 5-7: Measured signal wavelength as a function of crystal temperature and
tilting angle characterization for the 1 cm, 46.1 pm grating period Raicol crystal.
Tilting Angle and Temperature Dependence for
two pump wavelengths at 24*C and 40*C
(AdVR Crystal 46.20pm 2.5cm )
Temperature versus Wavelength
(AdVR Crystal 46.20pm 2.5cm)
1564
e
1562
1560
E
c
0
y = 0.0625x + 1559.3
1558
-+-776.205 nm pump
1556
-N-779.917 nm pump
y = 0.0688x + 1551.2
1554
1552
29
34
39
-U-776.205nm
temp=40.OC
4
0
0* 3
C
-*-779.917nm
temp=24.OC
0* 2
C
F
4
24
-+-776.205nm
temp=24.OC
6
0* 5
e
-"779.917nm
temp=40.OC
1
0
1552
1554
1556
1558
1560
1562
1564
Wavelength with max emission (nm)
Temperature of crystal (*C)
Figure 5-8: Measured signal wavelength as a function of crystal temperature and
tilting angle characterization for the 2.5 cm, 46.2 pm grating period AdVR crystal.
64
Using Sellmeier equations (Appendix A) allowed us to estimate the phase matching
condition by calculating the index of refraction of the pump, signal and idler, as a
function of temperature and wavelength. The grating period of the crystal for which
collinear emission would be obtained can then be calculated from the quasi-phase
matching condition. The temperature or the grating period can be fixed in order to
calculate where collinear degenerate emission can be obtained (Fig. 5-9). The model
can also be used for nondegenerate emission.
Degenerate Collinear Emission for Fixed
Grating Periods of the PPKTP crystal
y
E
>
Degenerate Collinear Emission as function of
Grating Period for Fixed Temperatures
1620
-0.2983x + 1576.3
1610
i561.0
E
-+*-20 deg C
-- 40 deg C
,ir- 60 deg C
-4-80 deg C
y = -0.2853x + 1590.5
1560.8
20
1580
1570
-*-46.2
-30
-
1590--+-46.1
-0-46.15
y = -0.31 6x
1600
70
120
0
Temperature (C)
1550
46.08 46.1 46.12 46.14 46.16 46.18 46.2
Grating Period (um)
Figure 5-9: Theoretical model for degenerate collinear emission for fixed temperatures
(a) and for fixed grating periods (b).
PPKTP crystals with different periods and lengths were modeled using the method
explained in Appendix A in order to find the combination of grating period and
operating temperature for which collinear emission (both degenerate case and nondegenerate case) at a specified wavelength was possible while accounting for variation in
crystal period and pump laser wavelength emission. This way, we knew that if there
was a deviation in the manufactured crystal grating period or if our pump wavelength
shifted, we would still be able to tune our crystal in either temperature or angle, or by
going to a slightly nondegenerate wavelength in order to still obtain collinear SPDC
emission.
65
5.3.2
Narrowband Filtering
Proper filtering of the pump beam is necessary to ensure that the light collected into
the output fiber is indeed the downconverted light and not pump leakage. Several
dichroic mirrors were placed in the setup, each transmitting only about 2% of the
pump power. Additionally, an Optigrate reflection volume Bragg grating was used to
filter out unwanted emissions and obtain a collinear degenerate emission at 1560.5 nm.
In order to select the proper filter bandwidth, a theoretical model of the SPDC
emission spectrum and Bragg grating profile was constructed. The SPDC emission
was approximated by a sinc-squared function with argument (AkL/2) [20]. At Ako =
0, the sinc function is at its maximum and the FWHM and zero-to-zero values can
be found by setting AkL/2 to i7/2 and ±7r respectively. At the center of the phasematching curve, the quasi-phase matching condition can be written as:
-
Ako = Ek - k9 - ki -
2w
= 0.
Ao
(5.6)
where A, is the grating period of the crystal and kP, k 8, ki our initial set of pump,
signal, and idler k-vectors. Fixing k., we want to find a k, and corresponding ki such
that:
Ak =-
- k2 - ki-
=
(5.7)
to yield the phase-matching bandwidth given by 2(A, - A,). In order to do so, we let
A = A, + AA such that the phase mismatch is exactly zero at A, or Aj:
'Ac
27r
= k, - ks - k-> k_+
Ao
A
= 0
=0,
A
(5.8)
(5.9)
and solving for Ak yields:
Ak =
A
66
AO
.
(5.10)
Substituting Eq. 5.10 into Eq. 5.7 we obtain the following relationship:
27r
2r
27r(A - A)
2,7rzA
r
A
AO
A Ao
AAo
L
(5.11)
Finally we find:
A2
A
(5.12)
2L
Using Excel solver we find A, and Ai for which the conditions above are satisfied
and the bandwidth can be easily estimated. We have also calculated the sinc-squared
phase-matching function as a function of the signal wavelength based on the Sellmeier
equations as shown in Fig. 5-10. We obtained a phase matching bandwidth of -1 nm
FWHM for the 3 cm crystal, in agreement with our estimate using Eq. 5.12.
1.0
0.8
0.6
0.4
0.2
1558
1560
1562
1564
Figure 5-10: Calculated SPDC emission for 3 cm crystal with a grating period of
46.1 pim.
SPDC bandwidth for crystals with different crystal lengths was calculated using the
steps above to calculate our theoretical Bragg grating efficiency. The Bragg grating
itself was modeled for different bandwidths, closely matching the specifications provided by the manufacturer.
We then combined our calculated SPDC emission and
Bragg grating models for different crystal lengths and Bragg grating bandwidth to
estimate the parameters for optimal efficiency of the SPDC emission, after reflection
from the Bragg grating as seen in Fig. 5-11.
67
1nm Filter
1.2nm Filter
1.4nm Filter
1.8nm Filter
2nm Filter
2.2nm Filter
Figure 5-11: Calculated SPDC emission with calculated Bragg filter shape for the
2.5 cm crystal (red curve), 3 cm crystal (green curve), and 5 cm crystal (blue curve).
The best compromise between performance and what could be manufactured by Optigrate was a Bragg filter with 1.8 nm bandwidth FWHM.
U
0.6
U
0.6
E
0
Z
0.4
0.2
1559
w
e60
161
156t
SPDC wavelength (nm)
Figure 5-12: 1.8 nm Optigrate Bragg filter efficiency normalized to total SPDC output: 97.54% for the 2.5 cm crystal (red curve), 97.78% for the 3 cm crystal (green
curve), 97.43% for the 5 cm crystal (blue curve).
68
Once the filters were received, the bandwidth was characterized experimentally by
using a tunable laser and scanning the wavelength from 1559 nm to 1562 nm at a fixed
angle of reflection. The FWHM bandwidth of the Bragg filters was experimentally
determined to be ~ 1.8 nm as expected with a filter shape given by the curve in
Fig. 5-13.
0.9
0.8
0.70
r. 0.6
0.5
0
0.4
0.3
;
0.2
0.1
0
1558.5
1559
1559.5
1560
1560.5
1561
1561.5
1562
1562.5
Wavelength (nm)
Figure 5-13: Bragg grating characterization at fixed angle varying the wavelength.
5.3.3
Focusing Configuration
Over the years, a number of papers have looked into the importance of focusing the
pump and how it affects the SPDC emission [17] [51]. In the first investigation on
the effect of pump focusing on nonlinear optical interactions in 1968 [17], Boyd and
Kleinman introduced the focusing parameter (
=
L/b where b =
2
ZR =
27rnwiw/A is
the confocal parameter (or twice the Raleigh range ZR) of the fundamental Gaussian
beam and L is the length of the crystal with refractive index n as shown in Fig. 514. It was found that the optimum configuration for classical-field three-wave mixing
corresponded to the case when ( = 2.84, the confocal parameters of all the beams
were equal ( , =
=
s), and the beam waist of all three beams was at the center of
the crystal [17].
69
z
L
Figure 5-14: Focusing geometry in the crystal with Rayleigh range
ZR,
crystal length
L, beam-waist radius wo, and focus offset zo.
A key efficiency when characterizing a source of entangled photons is the heralding
efficiency. This is the probability that, given the detection of one member of a SPDC
pair, the twin photon is also measured. In recent studies, it has been shown that
the tight focusing configuration described above is not optimum in terms of heralding
efficiency [12].
1
1
.N
100
0.9
0.8
.
C.)
0
0.8 -c
C 0.6
as
0
CD
0.7
0.4
10
0
1
C
C)
0.1
0.6
. 0.2
0.
0
0.01
'
'
'__
1
10
0.1
pump focus 4
0.01 k_
0.01
' 0.5
100
0.1
10
1
pump focus 4
100
Figure 5-15: (a) and (b) Simultaneous optimization of the total collection probability
Psj and the symmetric heralding ratio r,. In (a) the collection probability is plotted
along the left axis and the heralding ratio is plotted along the right axis [13].
70
In our experiment, the detectors used required single mode coupling therefore a
focused pump configuration was chosen as a starting point in order to optimize the
coupling efficiency rather than the heralding efficiency. Experimental data by other
groups [63] shows that with a highly focused pump such as ours we would expect to
couple -75%
of the SPDC emission into single mode fiber. The trade-off between
brightness and heralding ratio is shown in Fig. 5-15.
with (,
For a tightly focused pump,
~2.8, the SPDC coupling ratio is upwards of 80% however this focusing
configuration is not optimal in terms of heralding ratio as rsi drops to -75%.
the pump is unfocused more spatial modes are present.
When
The single mode coupling
selects only a specific mode thus dropping the overall coupling efficiency, however,
the heralding efficiency is higher.
5.3.4
Alignment of the Source
To ensure that a high quantum visibility is achieved, a high classical visibility must be
achieved. A fairly large portion of the time spent in this experiment was dedicated to
perfecting the alignment procedure of the source. Similarly to the experimental setups
described in Chapter 4, a probe laser at 1560 nm was used. First the horizontally
polarized probe beam was sent into the interferometer from the signal arm and coupled
into a single mode fiber at the output of the idler arm.
In order to achieve 92%
coupling of the probe light from one arm to the other, the aspheric lenses in front of the
fibers were mounted on translation stages and slowly adjusted by an iterative process
on both arms while also adjusting the last two mirrors of the coupling arms. Once high
coupling efficiency into the single-mode fiber was achieved for horizontally polarized
probe laser light, the polarization was rotated using the pump halfwave plate HWP
P1 to vertical to ensure that the same coupling efficiency was observed. If this was
not the case, then the interferometer needed to be adjusted for classical interference,
thus maximizing the spatial overlap of the counter-clockwise and clockwise beams.
To do so, the probe light was sent into the signal arm at 450 polarization to simulate
the SPDC emission and bi-directional pumping. Classical interference was measured
in the single-mode fiber on the idler side after the analyzer PBS with the halfwave
71
plate (HWP Idler2) set to 450 , connected to the Agilent InGaAs power sensor. By
carefully tuning the mirrors inside the Sagnac interferometer and the temperature
of the PPKTP crystal, we were able to achieve the desired maximum value of the
classical fringe visibility. With this method, a classical interference of 98.5% was
achieved.
The procedure above describes the technique used to optimize the alignment of
the Sagnac interferometer using a probe laser. The assumption was that if we could
then match the pump beam to the probe beam, then the SPDC would also follow the
same optimized path and a high level of entanglement would be achieved. To match
the pump to the probe, two techniques were used to adjust the pump beam without
touching any optical components that affect the signal or idler (probe) paths. First,
a Thorlabs beam profiler was used to overlap the pump beam and probe beam at
two different locations (one right at the crystal, and one about 50 cm away from the
crystal, using an additional lens to refocus the beam). By going back and forth and
adjusting the pump mirrors, one mirror for the near-position and another mirror for
the far-position, we were able to overlap the two beams visually within the resolution
of the beam profiler. Then, the second step of the procedure was to perform DFG
once again and make sure that the emission was maximized and coupled into the fiber.
Generally, a good overlap of the beams with the beam profiler resulted in a highefficiency DFG process and required minimum adjustments to the pump. The final
alignment was made by sending in the pump and looking at the SPDC in the single
mode fibers using the APDs. Adjusting the two mirrors in front of the analyzer PBS
and translating the lenses in the zoom lenses allowed us to optimize the coincidence
counts, thus insuring that the correct photon pairs were getting coupled into the fiber.
72
5.3.5
Flux Characterization
Before even attempting to couple the SPDC emission into a single mode fiber, we
needed to understand how much SPDC power we were expecting. In a previous exto 6xIO
periment using a 1 cm PPKTP with 0.6 nm bandwidth at 800 nm, -2
pairs/mW/s were generated. The lower bound of the range corresponded to a minimally closed aperture in front of the fiber for spatial filtering while the upper bound
of the range corresponded to a fully open aperture. Scaling the generation rate to
our operating parameters using [20] and [70], the upper bound estimated expected
generation rate for 40 mW of pump power was found to be ~9 million pairs/s corresponding to -1 pW of SPDC signal (or idler). We confirmed this emission power by
imaging the output of our crystal with an uncooled Goodrich indium gallium arsenide
(InGaAs) shortwave infrared (SWIR) camera (SU320HX-1.7RT).
1
0.9
0.8
0.9-
0.7
0.8 0.6
0.7-
0.6 0.5
0.5E 0.40
0.4
0.30.2-
0.3
0.10:
0
2
0.2
2
6
0.1
8
12
8
Pixels
Pixels
Figure 5-16: Spontaneous parametric down conversion observed with a Goodrich
InGaAs camera at 350 C. The emission was focused to match the pixel size FWHM
(30 tm).
73
Sending 40 mW of 780.2 nm pump counterclockwise and maintaining the temperature of the PPKTP crystal at 35'C resulted in an emission of -420 fW centered
at 1560.5 nm. This specific wavelength was obtained by peaking the power of the
emission imaged by the camera by tuning the Bragg grating, and then using a tunable Agilent laser to measure the wavelength for which the grating was optimized.
We then compared this with the amount of light coupled into a multimode fiber connected to an Agilent InGaAs power sensor (81634A) and saw comparable amount of
power (430 fW). This number corresponds to the singles in the idler arm, therefore
the total SPDC power would be twice this amount, or ~860 fW. Accounting for the
75% optical system efficiency, this would correspond to roughly 1 pW of SPDC generated at the crystal which is within the range of our estimate (corresponding to half
of the upper limit).
The expected system efficiencies for the signal and idler arms are summarized in
Table 5.1, where the optics transmission efficiency and detector efficiency were experimentally measured, the Bragg grating efficiency was calculated, and the coupling
efficiency was taken to be the optimum case of 75%, as discussed in Section 5.3.3.
Signal channel
Idler channel
Optics Transmission
76%
75%
Bragg Grating
96%
96%
Coupling Efficiency
75%
75%
Detector Efficiency
19%
21%
Overall System Efficiency
10.39%
11.34%
Table 5.1: Optical component transmissions and overall system efficiencies
From Table 5.1, we see that a system efficiency of -10% is expected. Using an estimate for the generation rate of 4.5 million pairs/s, we would then expect to have
450 000 singles/s in each channel.
For a duty cycle of 10% the expected num-
ber of singles detected is in the order of 45 000 singles/s in each arm.
In com-
parison, we measured in uni-directionally pumped configuration -30 000 singles/s
and -500 coincidences/s for a system efficiency of 1.64% and a duty cycle of 10%
74
at a pump power of 40mW. This measurement implies a pair generation rate of
3 x 104/1.64%/10% = 18 x 106 pairs/s, which is almost twice the upper range of our
estimate. This is reasonable because in the experiment the estimate was based on,
not all pairs were collected through the open aperture. The 18 x 106 pairs/s estimate
also suggests that the multimode power measurement might not have collected all
the mode pairs.
Another potential discrepancy is that the measured singles include some fluorescence output from the crystal or other components.
Fluorescence is unpaired
and therefore has the effect of making the unconditional detection efficiency appear
smaller, which in turn makes the estimate of the pair generation rate higher than the
actual rate. Further investigation is needed to sort out the source of the discrepancy.
The SPDC coupling efficiency into single-mode fiber versus multi-mode fiber was
measured using the Agilent power sensor, pumping the crystal counterclockwise, and
found to be at best 32% in the signal arm.
As discussed in Section 5.3.4, when
coupling the probe laser from one arm to the other a coupling efficiency of more than
90% was achieved. One cause for lower coupling efficiency when coupling the SPDC
emission could be that our pump focusing is not optimized to the probe focusing.
In the future, the pump focusing should be scanned using a zoom lens to optimize
the coupling for a given signal and idler single-mode coupling setup. Physically, the
pump focusing affects the spatial-mode distribution of the SPDC emission output
and hence the collection optics for single-mode fiber coupling. This behavior is the
focus of Bennink's paper [13] that calculates the single-mode collection and heralding
efficiencies as a function of the pump focusing and the collection optics.
75
5.4
Quality of Entanglement
First, to ensure that multi-pair events were not causing additional photon detections,
a measurement of signal and idler counts per second was performed for different
pump powers (Fig. 5-17) as well as a measurement of the coincidence counts per
second for different pump power (Fig. 5-18). Multi-pair events could occur when the
nonlinear crystal is pumped hard and more than one pair is simultaneously generated
through SPDC. If that is the case, the linear relationship between the singles counts
and pump power and coincidence counts and pump power becomes nonlinear. Our
measured data points fit very nicely with the theoretical linear fit, indicating that it
is unlikely multi-pairs were generated and affected the resulting coincidence counts.
Since afterpulsing is minimal at this operating temperature of -20'C the likelihood
of re-triggering of additional avalanches is low as well.
(
ca
35000
30000
n
25000
c
20000
0
15000
-4-Signal
10000
-U-Idler
o
J9
5000
0
0
0
10
20
30
40
Pump power (mW)
Figure 5-17: Measured singles counts with subtracted dark counts (21 000 dark
counts/sec and 9 250 dark counts/sec for detectors 1 and 2, respectively) for signal, idler as a function of pump power (mW) operating at -20'C with 10% duty
cycle.
76
400
3
350
.0
300
-' 250o 200
i-150
150
0 0
--
100
50
0
0
10
20
30
Coincidence
40
Pump power (mW)
Figure 5-18: Measured coincidence counts with subtracted dark counts (21 000 dark
counts/sec and 9 250 dark counts/sec for detectors 1 and 2, respectively) for signal,
idler as a function of pump power (mW) operating at -20'C with 10% duty cycle.
Once it was determined that our detection system was optimized, we were ready to
look at the quality of the polarization entanglement generated by our source. Pumping
the crystal bidirectionally with ~20 mW of pump on each side and optimizing for
maximum visibility, an average 12
000 singles/s were detected in each arm after
dark counts were subtracted, corresponding to -20 000 counts/sec for detector 1 and
~10 000 counts/sec for detector 2. As mentioned in Section 5.2 the dark count rate
is the average rate of registered counts without any incident light. The numbers of
single counts per second and coincidence counts per second for the horizontal/vertical
(HV) basis and anti-diagonal/diagonal (AD) basis are summarized in Table 5.2 after
subtraction of dark counts, while accidental coincidence counts were not subtracted.
This data set corresponds to the highest average visibility of 97.1% observed for this
Sagnac source, despite not having the highest observed number of coincidence counts.
77
Signal HWP
Idler HWP
Signal Singles
Idler Singles
Coincidence Counts
H
V
10 000
10 000
170
H
H
10 000
13000
3
V
H
12 000
14 000
180
V
V
13 000
10000
4
+45
+45
12 000
12 000
175
+45
-45
12 000
12 000
3
-45
-45
11 000
12 000
175
-45
+45
11 000
13000
3
Table 5.2: Measured singles and coincidences in two mutually unbiased bases.
The average number 175 coincidences/s was observed and corresponds to an average
system efficiency of 175/11 375 = 1.53% for the signal arm and 175/12 000 = 1.45%
for the idler arm.
This average system efficiency of 1.5% is about a factor of 7
off from the optimum system efficiency of 10%. Additional measurements yielded a
slightly higher system efficiency of 1.64% however the quality of the entanglement was
worse at 95%. The highest system efficiency was achieved by pumping the crystal
counterclockwise. An average of 450 coincidences/s sec for 22 000 singles/s in the
signal arm and 23 000 singles/s in the idler arm was observed, corresponding to
a system efficiency of -2%.
This is still a factor of 5 off from the optimum system
efficiency, however it gives us some insights to potential reasons for the low conditional
probability.
It is possible that we are not coupling the same part of the SPDC
emission in both arms and thus we are not observing a correlation between all of the
photons, due to the potential multi-spatial composition of the SPDC emission. If
different modes are collected, no coincidence is present. Optimizing the source for
counterclockwise pumping is easier than bi-directionally and more photons belonging
to the same pair could be collecting, thus improving our conditional probability. A
more thorough discussion is provided in Section 5.5.
78
The full fringe visibility graph was not recorded for the highest observed visibility.
In the case of an average visibility of 94.97% in the horizontal and vertical (HV)
basis and 94.99% in the diagonal and anti-diagonal (AD) basis without subtracting
accidentals the following fringe visibility was recorded:
250
i
ca
200
0.
cc
150
0
0
-+-AD basis
u 100
--.
HV basis
50
0
50
100
150
HWP Angle
200
(0)
Figure 5-19: Average fringe visibility recorded: 94.97% in the horizontal and vertical (HV) basis and 94.99% in the diagonal and anti-diagonal (AD) basis without
subtracting accidentals.
These fringes were observed maintaining the idler HWP at a fixed angle and
scanning the signal HWP for both the HV and AD basis. The small difference in the
amplitudes of the fringes could be explained by pumping the crystal with a beam with
uneven amplitude in the H and V polarization components. Also, the dual wavelength
PBS had reflection and transmission efficiencies of 92% and 98% further complicating
the pump's ideal polarization. Upon additional optimization of the interferometer
classical interference with the probe, pump and probe alignment, pump polarization,
and coupling, we hope to raise this visibility to more than 99%, as seen in another
Sagnac source in our group [30].
79
Another small improvement in visibility can be made by subtracting accidental
coincidence counts. Accidental coincidences can occur for different reasons: when
a dark count photon arrives at the same time as another dark count, or when a
multi-pair event occurs and two photons that are not generated in the same pair
arrive simultaneously (within the coincidence window), or when a dark count photon
arrives at the same time as a generated photon, etc. The probability of getting an
accidental count per gate for detector 1 (P1 ) and detector 2 (P 2 ) can be calculated
from the following equations:
P1
P1
=
C111375
fg
628.5 x 106
=
1.81
x
10-
(5.13)
12000
628.5 x 106
=
1.91
x
10--
(5.14)
2
fg
where C 1 is the total number of counts per gate frequency
f9
and C2 is the total
number of counts per gate frequency fg. If these two events are independent, the
total accidental coincidences counts per second can be calculated:
C1
C2 )
f9
f9
g=(di + aol
()
(2)()
(2
+ a," + ...)(d2 + a2 + a2 + ... )5
628.5 x 106
where 628.5 MHz is the synthesizer frequency, di is the number of dark counts per
second from detector 1, d 2 is the number of dark counts per second from detector
2, a( 1 and a)
are the number of accidental counts per second between single pairs
generated detected at detector 1 and 2 respectively, a, ) and a)
are the number of
accidental counts per second between first order multi-pairs generated detected at
detector 1 and 2 respectively, and so forth. From Eq. 5.15, the average number of
accidental coincidence counts per second is therefore:
(11375)(12000) <
628.5 x 106
(5.16)
This number is confirmed when time shifting the arrivals of the photon pairs as less
than 1 coincidence per second was observed.
80
5.5
Discussion
A number of source parameters still need to be optimized in this experimental setup
in order to obtain a truly high-flux high-quality polarization entangled source. While
the highest average observed visibility of 97.1% is on par with previous source if not
higher, as discussed in Chapter 3, it is not as good as more recent measurements
in Sagnac interferometers with higher than 99% visibility [28].
The slightly lower
visibility could simply be attributed to the need for additional optimization of the
pump polarization. Additionally our tight focusing pump configuration requires more
accurate alignment than a collimated pump configuration.
A bigger source of concern is that the average system efficiency of 1.5% is about a
factor of 7 off from the optimum system efficiency of 10%. While a very high heralding
efficiency was not expected with our tight focusing configuration, a coupling efficiency
of ~75% was expected [13]. We estimated that the coupling efficiency of the SPDC
into single mode fiber was only -30%. Single mode fiber coupling of SPDC is not an
easy task even when pumping the crystal in one direction since the core of the fiber
is only ~ 5pm in radius. It is even more difficult to optimize both the clockwise and
counterclock-wise emissions into a single-mode fiber simultaneously when the crystal
is bi-directionally pumped because the two emissions can be slightly non-overlapping
which results in different parts of the SPDC getting coupled into the fiber.
Furthermore, the shape of the mode of the SPDC emission may not be optimal
for the coupling setup since the mode matching configuration is optimized using the
laser probe light. Pumping the crystal counterclockwise, the SPDC coupling efficiency
into single mode fiber versus multi-mode fiber was measured using the Agilent power
sensor and found to be at best 32% in the signal arm. A potential reason for the
lower coupling efficiency could be that the zoom lenses on the signal and idler arm
were simply not sufficiently optimized for the SPDC emission. We recently placed
the zoom lenses on translational stages for more accurate tuning.
Additionally, even the slightest difference in wavelengths between the signal and
idler results in a different propagation angle due to the different indices of refraction
81
inside the crystal for different wavelengths. Although the parallelism of the surfaces
of our PPKTP crystal was tested by sending the probe light at normal incidence and
trying to observe the path of the reflected beam, it was very difficult to precisely
determine how parallel and smooth the surface of the crystal was. If the facets of the
crystal are not parallel, it is much harder to overlap the clockwise and counterclockwise emission, in addition to all the other factors.
However, as discussed in Section 5.3.4, when coupling the probe laser from one
arm to the other a coupling efficiency of more than 90% was achieved by adjusting
the zoom lenses. Therefore we believe that additional factors were involved in the
reason of this low coupling efficiency. One other possible cause for lower coupling
efficiency was thought to be that our pump focusing was not optimized to the probe
focusing.
The biggest issue in terms of coupling, it seems, is that when coupling the SPDC,
we do not know exactly what the mode looks like. Tight focusing, careful temperature
tuning, the use of the Bragg grating, and the use of a single mode fiber helps us in
the understanding of the number of modes, collinear behavior, wavelength, and size
of our emission but does not guarantee high coupling efficiency. The collection of the
SPDC emission needs to be investigated further in our Sagnac source. It is possible
that our pump was too tightly focused.
Finally, one potential application discussed for this Sagnac source was to use it
in conjunction with Lincoln Laboratory's superconducting nanowire single photon
detectors to attempt to close the detection loophole (Appendix B). Currently our
source efficiency would not allow us to perform such an experiment however, by
defocusing the pump, and using the larger core single-mode fiber recently developed,
we hope to obtain better results. When the pump is defocused more spatial modes
of SPDC are generated. While the single-mode fiber gets rid of the extra modes,
thus lowering the overall coupling efficiency, a very high heralding efficiency may be
obtained by pumping the crystal harder, making it a suitable source for applications
such as loophole free detection.
82
Chapter 6
Conclusion
Polarization-entangled photons are essential resources for many quantum information
science applications, such as quantum key distribution and linear optics quantum computing. Compact, high-flux sources of high-quality polarization-entangled photons are
desirable for implementing these applications. In this thesis, we demonstrated our
ability to design and build sources of entangled light in different configurations.
The main goal of this thesis was to design a high-quality high-flux source of singlemode fiber-coupled polarization entangled photons at 1.56 pm. The high-flux aspect
of the source still needs to be improved. At this time, we believe that further optimization of the mode matching of the SPDC and coupling into single mode should
improve our 2% system efficiency closer to the expected 10% level. The 97.1% quality
of the polarization entanglement of our source could still be improved when comparing to the best Sagnac sources built, but it is still on par or better than other sources
of polarization entangled photons. All other criteria of the source were met and once
the efficiency is improved we plan to connect this source to larger core Superconducting Nanowire Single Photon Detector (SNSPD) at Lincoln Labs, further increasing
our system efficiency. These recent advancements in detection efficiency could mean
that we are closer than ever to achieving a loophole free violation of Bell's Inequality,
which could be a potential application for our Sagnac source once it is fully optimized
(Appendix B).
83
In addition, over the course of this thesis work, a linear Mach-Zehnder source
for the teaching laboratory was developed. The novelty of the source came from its
compact and versatile design allowing students to perform fairly easily fundamental
quantum optics experiments such as creating post-selected polarization entanglement,
measuring an HOM dip and performing a CHSH inequality violation. A quantum
visibility of ~90% demonstrated reasonably high-quality entanglement.
An HOM
dip with a two-photon quantum interference visibility of 80% was also demonstrated.
Going from generating entangled photons to measuring the HOM dip required very
few steps - rotating halfwave plates and translating a prism on a stage - confirming
the versatility and ease of use of our system.
84
Appendix A
Calculation of Crystal Grating
Period using Sellmeier Coefficients
The Sellmeier equations are an empirical relationship between refractive index and
wavelength for a particular transparent medium [43].
The equations are used to
determine the dispersion of light in the medium.
Index
A
B
C
D
n(y)
2.09930
0.92268
0.04677
0.01384
n(z)
2.12725
1.18431
0.05149
0.66030
E
F
100.00507
0.00969
Table A.1: Table of Sellmeier coefficients used for all phase matching calculations
The indexes of refractions at room temperature T= 25 0 C for different wavelengths
and polarizations are calculated using the Sellmeier coefficient above:
B
n(y) =
(AY +
n(z) =
(Az + 1 -
1 - Cy/A 2
2
(A.1)
- DY x A2 )
+
1
-
z
zA
- Fz x A2
(A.2)
For example, the indices of refraction of a pump wavelength of 780 nm and signal
and idler wavelength of 1560 nm are calculated at room temperature to be :
85
n(y)
n(z)
pump A,
1.75728
1.84583
signal A,
1.73336
1.81495
idler Ai
1.73336
1.81495
Table A.2: Calculated indices of refraction at T= 25'C
The grating period of the crystal at room temperature T= 25'C is then calculated
from the quasi phase matching condition:
1
A =_
(A.3)
(n(y),/A, - n(y),/As - n(z)i/Ai)
Since our x-propagating PPKTP crystal has type-II emission, the signal and idler
have orthogonal polarizations. For the horizontal polarized pump and signal the index of refraction is taken along the y axis (n(y)) and for the vertically polarized idler
the index of refraction is taken long the z axis (n(z)).
When varying the temperature away from T= 25'C by AT, the difference in the
index of refraction An is found to be [38]:
An(y) = (0.1997/A 3 - 0.4063/A 2 + 0.5154/A + 0.5425) x (10-5)AT
(A.4)
An(z) = (-0.5523/A + 0.3424A 2
(A.5)
-
1.7101A + 3.39) x (10~5)AT
Therefore the new indices of refraction at temperature T are n(y)' = n(y)-+An(y) and
n(z)'
=
n(z)
+ An(z). Additionally, when calculating the grating period for different
temperatures, the thermal coefficient of expansion of the crystal must be included in
the calculation. The thermal expansion coefficients of PPKTP are a
=
6.7*10-6 and
3 = 11 * 10-9. The grating period of the crystal then becomes:
A=
11
x (1 + a(T - 25) + #(T
(n'(y),/A, -- n'(y)s/A, - n'(z)i/Ai)
86
-
25))
(A.6)
Appendix B
Loophole-Free Violation of Bell's
Inequality
In 1935, Albert Einstein, Boris Podolsky, and Nathan Rosen published a paper in
which they introduced a thought experiment (now referred to as EPR) which was a
challenge to the Copenhagen Interpretation of quantum mechanics [26]. First they
defined a complete physical theory as one in which every element of the physical reality
must have a counterpart in a complete physical theory. They defined an element of
reality as satisfying a sufficient criterion: if without an any was disturbing a system
we can predict with probability equal to one the value of a physical quantity, then
there exists an element of the physical reality corresponding to this physical quantity.
This physical quantity would be a hidden variable. In 1952, David Bohm published
a suggested interpretation of the quantum theory in terms of such hidden variables.
However in 1966, John S. Bell published a new interpretation of the EPR paradox
in his breakthrough paper on the problem of local hidden variables in quantum mechanics [6]. In his paper, Bell started from essentially the same assumptions as did
EPR, 1) reality (microscopic objects have real properties determining the outcomes
of quantum mechanical measurements) and 2) locality (reality is not influenced by
measurements simultaneously performed at a large distance). Bell was able to derive
from these assumptions an important result, the Bell's inequality, the violation of
which by quantum mechanics implying that at least one of the assumptions must be
87
abandoned if the experiment would turn out to satisfy quantum mechanics, which
was later demonstrated in the experiment proposed by Clauser, Horne, Shimony and
Holt [23].
Over the past thirty or so years, a great number of Bell test experiments have been
successfully conducted. The first experiment was performed by Stuart Freedman and
John Clauser himself in 1972
[31].
This experiment provided the first confirmation
that quantum mechanics is non-local. However this Bell test used Freedmans inequality, a variant of the CH74 inequality [22], which is itself a variant of the original CHSH
inequality. It wasnt until 1981, when Alain Aspect and his team conducted three Bell
tests using calcium cascade sources [4] [3] [2] that the actual CHSH inequality was
used and thus this particular test is considered the first experimental application [3].
Many other Bell test experiments have been carried out since. Paul Kwiat's high
efficiency source demonstrated one of the largest violations of Bells inequality [47]
Nicholas Gisin's group made a Bell test experiment that showed that distance did
not destroy entanglement [65]. At the same time, Anton Zeilinger's team conducted
an experiment where the choice of the detector was made using a quantum process
to ensure that it was random, violating further than ever before Bell's inequality [66].
Although the series of increasingly sophisticated Bell test experiments has convinced
the physics community in general that local realism is untenable, critics would point
out that the outcome of every single experiment that violates a Bell inequality could,
at least theoretically, be explained by faults in the experimental setup, experimental procedure or that the equipment used did not behave as well as it is supposed
to. It became apparent that these experiments were all subject to assumptions, the
so-called "loopholes" of Bells inequality.
Until recently most experimental Bell tests used optical setups in order to separate
at enough distance the entangled particles and ensure that the measurement of each
was not causally connected to the other (ensuring that even light-speed interactions
between them cannot occur). Low loss optical fibers permit even longer distribution
of prepared entangled states, thus ruling out a locality loophole. However low detection efficiency does not ensure a local realistic model which in turn yielded a detector
88
efficiency loophole. In this assumption, it could be possible to "engineer" quantum
correlations (the experimental result) by letting detection be dependent on a combination of local hidden variables and detector setting, since particles are not always
detected in both arms of the experiment. In other words it refers to the question of
whether or not "fair sampling" (the sample of detected pairs is representative of the
total pairs emitted regardless of the efficiency) is a correct assumption. This loophole
was closed recently in an experiment using trapped ions [591, which allowed a much
higher measurement efficiency of the ion states. Unfortunately, because the ions have
to be held very closely together inside a single trap, this experiment does not satisfy
the condition of separation necessary for a loophole free Bell test.
At this time, no loophole free Bell test has been successfully carried out, despite
experimenters stating that loophole free Bell tests are possible [46]. Recent advancements in technology have led to significant improvement in the efficiencies of new
superconducting detectors such as Superconducting Nanowire Single Photon Detector (SNSPD) and Transition Edge Sensor (TES) [36], thus we could argue that we
are closer than ever to achieving a loophole free violation, if indeed it is possible.
SNSPDs perform single-photon counting with exceptional sensitivity and time resolution at near-infrared wavelengths.
Recent advances in SNSPDs allow operation
at photon detection rates exceeding 100 MCounts/s with record-high detection efficiency. Detectors with active areas > 14pm in diameter have been demonstrated
with -80% device detection efficiency at 1550 nm wavelength, enabling multi-mode
optical coupling for free-space applications [24].
89
90
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